Unified Systems of FB-SPDEs/FB-SDEs with Jumps/Skew Reflections and Stochastic Differential Games

Uniﬁed Systems of FB-SPDEs/FB- S DEs with Jumps/Sk ew Reﬂections and Sto c hastic Diﬀeren tial Games 1 W an yan g Dai 2 Department of Mathematics and State K ey Labo ratory of Novel Softw ar e T ec hnology Nanjing Universit y , Na njing 210 093, China Email: nan5lu8@netra.nju.edu.cn Date: 14 September 20 15 Abstract W e study four systems and their in teractions. First, w e for mulate a uniﬁed system of coupled forward-backw ar d sto chastic p artial diﬀerent ial equations (FB-SPDEs) with L´ evy jumps, whose drift, diﬀusion, and jump co eﬃcients ma y in volve par tial diﬀeren- tial oper ators. A solution to the FB-SPDEs is deﬁned by a 4- tuple ge neral dimensional random vector-ﬁeld pro cess ev o lving in time together with p ositio n parameters ov er a domain (e.g., a hyperb ox or a manifold). Under an inﬁnite seq uence o f generalized loc a l linear growth and Lipsc hitz conditions, the well-p osedness of an adapted 4-tuple strong solution is pr ov ed ov er a suitably co nstructed top ologica l s pace. Second, we consider a uniﬁed system of FB- SDEs, a sp ecial fo rm of the FB-SPDEs, howev er , with skew bo und- ary reﬂections. Under randomized linear gr owth and Lipschitz conditions together with a general co mpletely - S condition on r e ﬂections, we prove the well-po sedness of an adapted 6-tuple weak solution with b oundary r e gulators to the FB-SDEs by the Skoroho d problem and an o scillation inequality . Particularly , if the sp ectral radii in some sense for reﬂec tion matrices are s trictly less than the unity , a n a dapted 6-tuple stro ng solution is concerned. Third, w e for mulate a sto chastic diﬀeren tial game (SDG) with general n umber of play- ers based on the FB-SDEs. By a solution to the FB-SPDEs, we get a solution to the FB-SDEs under a given c ontrol rule and then obtain a Pareto optimal Nash equilibrium po licy pr o cess to the SDG. F ourth, we study the a pplications of the FB-SPDEs/ FB-SDEs in queueing systems a nd quantum s tatistics while we use them to motiv ate the SDG. Key w ords and phrases: sto chastic (partial/ordinary) diﬀerential equation, L´ evy jump, skew r eﬂection, completely- S condition, Skoroho d problem, o scillation inequal- it y , s to chastic diﬀeren tial game, Pareto o ptimal Nash e q uilibrium, queueing net work Mathematics Sub ject Class iﬁcation 2000: 60H15, 60H10 , 91A1 5, 91 A2 3, 60 K25 1 This work was presented in parts or as a whole at 8th W orld Congress in Probabilit y and S tatistics, Istan- bul (2012), American I nstitute of Mathematical Sciences A n nual C onference, Madrid (2014), 8th International Congress of Industrial and Applied Mathematics, Beijing (2015), and at conferences in Suzhou (2011), Min- neap olis (2013), Chengdu (2013), Sh anghai (2014), Jinan (2014), Nanjing (2014), Beijing (2015), Chongqing (2015), Changsha (2015), Kunming (2015), S heny ang (2015), etc. The p resenta tions as invited plenary talks in SCETs 2014/2015 and I CPDE 2015 w ere video-tap ed by W anfa ng Data and can be downloaded online. The author thanks the helpful commen ts from the participants. An earlier version on the uniﬁed B-SPDE w as p osted v ia arxiv in May of 2011 and on the un iﬁed systems of FB-SPDEs/FB-SDEs with skew reﬂections w as p osted via arxiv on June 16 of 2015. 2 Supp orted by National Natural Science F oundation of China with Grant No. 10971249 and Grant No. 1137101 0. 1 1 In tro duction W e stu dy four systems and their in teractions: a un iﬁed system of coupled forw ard -bac kward sto c h astic partial diﬀerential e quations (FB-SPDEs) with L ´ evy jumps; a uniﬁed system of FB-SDEs, a sp ecial form of the FB-SPDEs, ho we v er, with skew reﬂections; a sto chastic diﬀeren tial game (SDG) problem with general n umber of pla y ers based on the FB-SDEs; and a system of queues and their asso ciated reﬂecting diﬀusion approximat ions. More pr ecisely , there are four inte rconnected and streamlined aims in volv ed in our discussions. The ﬁrst aim is t o study the adapted 4-tuple strong solution ( U, V , ¯ V , ˜ V ) to the uni- ﬁed system of coupled FB-SPDEs w ith L´ evy jum p s with resp ect to time-p osition parameter ( t, x ) ∈ R + × D ,                  U ( t, x ) = G ( x ) + R t 0 L ( s − , x, U, V , ¯ V , ˜ V ) ds + R t 0 J ( s − , x, U, V , ¯ V , ˜ V ) dW ( s ) + R t 0 R Z h I ( s − , x, U, V , ¯ V , ˜ V , z ) ˜ N ( λds, dz ) , V ( t, x ) = H ( x ) + R τ t ¯ L ( s − , x, U, V , ¯ V , ˜ V ) ds + R τ t ¯ J ( s − , x, U, V , ¯ V , ˜ V ) dW ( s ) + R τ t R Z h ¯ I ( s − , x, U, V , ¯ V , ˜ V , z ) ˜ N ( λds, dz ) , (1.1) where, t ∈ [0 , τ ] and τ ∈ [0 , T ] is a stoppin g time with regard to a ﬁ ltration deﬁn ed later in the pap er, Z h = R h − { 0 } or R h + for a p ositive integ er h , and s − denotes the corresp ond ing left limit at time p oint s . I n particular, D ∈ R p with a giv en p ∈ N = { 1 , 2 , ... } is a connected domain, for examples, a p -d imensional b o x , a p -d im en sional ball (or a general manifold), a p -dimensional sph ere (or a general Riemannian manifold), or the whole Euclidean sp ace R p of real n umb ers itself. The F- SPDE in (1.1) is with the g iv en initial random vect or-ﬁeld G , while the B-SPDE in (1.1) has the kno wn terminal rand om v ector-ﬁeld H . In (1.1), U and V are r -d im en sional and q -dimensional random v ector-ﬁeld pro cesses resp ectiv ely , W is a s tand ard d -dimensional B ro wnian motio n, and ˜ N is a h - dimensional ce nt ered L ´ evy jump p ro cess (or cente red sub ord inator). F urtherm ore, th e p artial diﬀeren tial op erators of r -dimensional v ector L , r × d -dimensional matrix J , and r × h -dimensional m atrix I are functionals of U , V , ¯ V , ˜ V , and their partial d eriv ativ es of up to the k th order for k ∈ { 0 , 1 , 2 , 3 , ... } . S o do the partial diﬀerenti al o p er ators of q -dimensional vec tor ¯ L , q × d -dimensional matrix ¯ J , and q × h -dimensional matrix ¯ I . More precisely , for eac h A ∈ {L , J , ¯ L , ¯ J } , A ( s, x, U, V , ¯ V , ˜ V ) ≡ A ( s, x, ( U, ∂ U ∂ x 1 , ..., ∂ k U ∂ x i 1 1 ...∂ x i p p )( s, x ) , (1.2) ( V , ∂ V ∂ x 1 , ..., ∂ k V ∂ x i 1 1 ...∂ x i p p )( s, x ) , ( ¯ V , ∂ ¯ V ∂ x 1 , ..., ∂ k ¯ V ∂ x i 1 1 ...∂ x i p p )( s, x ) , 2 ( ˜ V , ∂ ˜ V ∂ x 1 , ..., ∂ k ˜ V ∂ x i 1 1 ...∂ x i p p )( s, x, · ) , · ) , where the dot “ · ” in ˜ V ( s, x, · ) and its asso ciated partial deriv ativ es denotes the inte gration in terms of the so-called L´ e vy m easur e. How ev er, if A ∈ {I , ¯ I } , the last lin e on the right-hand side of (1.2) should b e c hanged to the form, ( ˜ V , ∂ ˜ V ∂ x 1 , ..., ∂ k ˜ V ∂ x i 1 1 ...∂ x i p p )( s, x, z ) , z , · ) . Figure 1: Sample Surfac e Solution to the FB-SPDEs Note that our partial diﬀeren tial op erators presen ted in (1.2) can b e general-nonlinear and high-order, e.g., A ( s, x, U, V , ¯ V , ˜ V ) = f ( ∂ k V ( t, x ) ∂ x i 1 1 ...∂ x i p p ) for a general nonlinear functional f , where r, i 1 , ..., i p are n onnegativ e intege rs satisfying i 1 + ... + i p = k with k ∈ { 0 , 1 , 2 , 3 , ... } . F urthermore, the initial rand om vec tor-ﬁeld G , the terminal random v ector-ﬁled H , and the 4-tuple solution pro cess ( U, V , ¯ V , ˜ V ) can b e complex-v alued. Under an inﬁn ite s equence of generalized lo cal linear gro w th an d Lipsc hitz conditions, we pro v e th e existence and un iqueness of an adapted 4-tuple s tr ong solution to the FB-SPDEs in a su itably constructed fu nctional top ologica l space. The solution to the un iﬁed sys tem in (1.1) can b e in terp reted in a sample sur face m anner with time-p osition parameter ( t, x ) (see, e.g., V ( t, x ) in Figure 1 for such an example). The quite inv olv ed tec hn ical p ro of dev elop ed 3 in this paper is extended from our earlier w ork su mmarized in Da i [17 ] (arxiv, 2011 ) for a uniﬁed B-SPDE. More p r ecisely , the newly uniﬁ ed system of coupled FB-SPDEs in (1.1) co v ers many existing forward and/or bac kward SDEs/SPDEs as sp ecial cases, where the partial diﬀer- en tial op erators are tak en to b e sp ecial form s . F or examples, sp eciﬁc single-dimensional strongly n onlinear F-SPDE and B-SPDE dr iv en solely by Bro wn ian motio ns can b e r esp ec- tiv ely deriv ed for the pu rp ose of optimal-utilit y b ased p ortfolio choic e (see, e.g, Musiela and Zariphop oulou [38]). Here, the strong nonlinearit y is in t he sense addressed b y Lions and Souganidis [34] an d Pardoux [42]. F urthermore, the single-dimensional sto chastic Hamilton- Jacobi-Bellma n (HJB) equations are also examples of our u niﬁed system in (1.1), whic h are s p eciﬁc B-SPDEs (see, e.g ., ∅ ksendal et a l. [4 1] and references therein). Note that the pro of of the we ll-p osedness concerning solution to the B-SP DE deriv ed in Musiela and Z a- riphop oulou [38 ] and solution to the HJB equation deriv ed in ∅ ksendal et al. [41] is co vered b y the stu dy in Dai [17] (arxiv, 2011) although the authors in b oth [38] and ∅ ksendal et al. [41] claim it as an op en p roblem. Th e pro of of the well-posedn ess ab out solution to th e F-SPDE deriv ed in Mus iela and Zariphop oulou [38] is co v ered b y the ev en more uniﬁed discussion for the coupled FB-SPDEs in (1.1) of this pap er. Actually , partial motiv ations to enhance the uniﬁed B-SPDE in Dai [17] (arxiv, 2 011) to the coupled FB-SPDEs in (1.1) are from the conference discus sion [22] during 45 minutes in vited talk presented by Zariphop oulou in ICM 2014, w h ere th e current author claimed that the w ell-p osedness of solution to the F-SPDE in [38] can b e pro v ed by the metho d dev elop ed in Dai [17] (arxiv, 2011). Besides these existing examples, our motiv ations to study the coupled FB-SPDEs in (1.1) are also fr om optimal p ortfolio managemen t in ﬁnance (see, e.g., Dai [16, 20]), and multi-c hannel (or m u lti-v alued) image regularizatio n suc h as color images in computer vision and net wo rk app lications (see, e.g., Caselles et al. [7]). In this part, we also show the usages of our uniﬁed system in (1.1) in heat diﬀusions and quan tu m Hall/anomalous Ha ll eﬀects as t wo illustrativ e exa mples to supp ort our ﬁrst aim. Mathematically , we reﬁne a stochastic Diric hlet-Po isson problem from heat diﬀu sions and use sto chastic Schr ¨ o dinger equation as mod el for Hall eﬀects in quantum statistics. It is worth to p oint out that the proving metho d ology dev elop ed in the current p ap er and its early ve rsion in Dai [17] (arxiv, 2011) is aimed to p ro vide a general theory and fr amew ork to show the w ell-p osedness o f a uniﬁed general system class of t he c oupled FB-SPDEs in (1.1). Ho we v er, some sp eciﬁc forms of the FB-SPDEs in (1.1) (either in forwa rd mann er or in bac k ward manner) ma y b e solv ed b y alternativ e tec hniqu es, e.g., the author in h is Fields Metal a w arded w ork (Hairer [2 6] and ICM 20 14) s olv es the KPZ equation by rough path tec hn ology , and f urthermore, th e related rough path theory can deal with the lac k of either temp oral or s p atial regularity (see, e.g., Hairer [26] and r eference therein). The second aim of th e pap er is to pr o ve the well- p osedn ess of an adapted 6- tuple weak solution (( X, Y ) , ( V , ¯ V , ˜ V , F )) with 2-tuple b ound ary regulator ( Y , F ) to the (p ossib le) non- Mark ovia n system of coupled FB-SDEs with L ´ evy ju mps and sk ew reﬂections under a give n 4 con trol rule u ,                                                               X ( t ) = b ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , · ) , u ( t − , X ( t − ) , · ) dt + σ ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , · ) , u ( t − , X ( t − )) , · ) dW ( t ) + R Z h η ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , z ) , u ( t − , X ( t − )) , z , · ) ˜ N ( dt, dz ) + Rd Y ( t ) , X (0) = x, Y i ( t ) = R t 0 I D i ( X ( s )) d Y i ( s );                  V ( t ) = c ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , · ) , u ( t − , X ( t − ) , · ) dt − α ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , · ) , u ( t − , X ( t − )) , · ) dW ( t ) − R Z h ζ ( t − , X ( t − ) , V ( t − ) , ¯ V ( t − ) , ˜ V ( t − , z ) , u ( t − , X ( t − )) , z , · ) ˜ N ( dt, dz ) − S dF ( t ) , V ( T ) = H ( X ( T ) , · ) , F i ( t ) = R t 0 I ¯ D i ( V ( s )) dF i ( s ) . (1.3) In (1.3), X is a p -dimensional p r o cess go verned by the F -SDE with sk ew reﬂectio n matrix R and V is a q -dimensional pro cess go v erned by the B-SDE with skew reﬂ ection matrix S . F urtherm ore, Y can increase only when X is on a b oundary D i with i ∈ { 1 , ..., b } and F can increase only wh en V is on a b oundary ¯ D i with ∈ { 1 , ..., ¯ b } , where b and ¯ b are t w o nonn egativ e in tegers. Both Y and F are the regulating pro cesses with p ossible ju mps to p ush X a nd V bac k into t he state space s D and ¯ D resp ectiv ely . They are parts of the 6 -tuple solution to (1.3) a nd dete rmined b y solution pairs to the well-kno wn Sk oroho d problem (see, e.g., Dai [14], Dai an d Dai [12 ], or Section 6 of the curren t p ap er for suc h a deﬁnition). Th u s, w e call them as Sk oroho d regulators (see, Figure 2 for suc h an example). Note that, comparing with the un iﬁ ed s ystem in (1.1), th e co eﬃcien ts app eared in (1.3) do not con tain any partial deriv ativ e op erator but the FB-SDEs themselv es inv olv e ske w b oundary reﬂections. The pro of for th e w ell-p osedness of an adapted 6-tuple w eak solution to the FB-SDEs is based on t wo general conditions. The ﬁ rst one is a general completely- S condition (see, e.g., Dai [14], Dai and Da i [12], and Figure 2 for an illustr ation). The non-uniqueness of solution to an asso ciated Sk oroho d problem und er this condition is one of the ma jor diﬃculties in the pro of. Th e second one is the generalized lin ear growth and L ipsc hitz cond itions, where the con ve nt ional gro w th and Lipsc h itz constan t is replaced by a p ossible un b ounded but mean-squarely inte grable adap ted sto c h astic pro cess (see, e.g., Dai [16, 20]). In particular, if the completely- S condition b ecomes more strict, e.g ., w ith additional requirements that the sp ectral radii in certain sense for b oth reﬂectio n matrices are strictly less than the unit y , a unique ad ap ted 6-tuple strong solution will b e concerned . Concerning coupled FB -SDEs, it motiv ates a hot resea rc h area (see, e.g., ∅ ksend al et al. [41] ab out the discussion of coupled FB-SDEs w ith no b ound ary r eﬂection, K aratzas and Li [31] ab out the study of Brownian motion drive n B-SDE with reﬂ ection, and r eferences therein). Ho w ev er, to our b est kn o w ledge, the c oupled system in (1. 3) with doub le ske w reﬂection matrices and th e well-posedness stu dy in terms of an adapted 6-tuple w eak s olution with L ´ evy jumps and un der a general completely- S cond ition throu gh the Sk oroho d problem 5 Figure 2: Skew and Inw ard Reﬂection with Sk oroho d Regulator under Co mpletely - S Condition are new and for the ﬁrst time in this area. The third aim of the pap er inv olv es tw o folds. On the one hand , we use the 4-tuple solution to the coup led FB -SPDEs in (1.1) to obtain an adapted 6-t uple solution to the system in (1.3); O n the other hand , we u se th e obtained adapted 6-tuple solution to determin e a P areto optimal Nash equilibrium p olicy pro cess to a n on -zero-sum SDG p roblem in (1.4), whic h is newly form u lated b y the FB-SDEs in (1.3). In this game, there are q -pla y ers and eac h p la yer l ∈ { 1 , ..., q } has his own v alue function V u l sub j ect to the system in (1.3 ) u nder an admissible con trol p olicy u . Every pla yer l c ho oses an optimal p olicy to maximize his own v alue function o ver an admissible p olicy set C while the summation of all v alue fu nctions is also maximized, i.e., sup u ∈C V u l (0) = V u ∗ l (0) (1.4) for eac h l ∈ { 0 , 1 , ..., q }} , w here, V u 0 ( t ) = q X l =1 V u l ( t ) . (1.5) Note that th e total v alue function V u 0 (0) do es not ha ve to b e a constant (e.g., zero), o r in other w ords, the game is not necessarily a zero-sum one. The con tribu tion and literature review of the stud y asso ciated with the game in (1.4)-(1.5 ) for the third aim can b e summarized as follo w s. One of the imp ortan t solution metho d s for SDE based optimal control is the dyn amic programming. I n general, this metho d is related to a sp ecial case of the un iﬁed system in (1.1) (or its earlier u niﬁed B-SP DE form in Dai [17] (arxiv, 2011)), e.g., the sp eciﬁc B-SPDE with q = 1 ( called sto c h astic HJB equation) in 6 P eng [44] with no ju m ps and ∅ ksendal et al. [41] with jump s. Here, we extend the discu s sions in P eng [44] and ∅ ksendal et al. [41] to a system of generalized coupled forw ard-bac kwa rd orien ted stoc hastic HJB equations w ith jumps corresp onding to the case that q > 1. More imp ortant ly , this system pro vides an eﬀectiv e w ay to r esolve a non-zero-sum SDG problem with jumps and general num b er of q pla yers, whic h sub jects to a non-Marko vian system of coupled FB-SDEs with L ´ evy jumps and sk ew reﬂections (see, e.g., Figure 3 for such a game platform (partially adapted from Dai [15])). By a solution to the FB-SPDEs in (1.1), we Figure 3: A 5-player game platform based on brain and s atellite communication determine a solution to the FB-SDEs in (1.3) under a giv en con trol r ule and then obtain a P areto optimal Nash equilibrium p olicy pro cess to the non-zero-sum S DG pr oblem in (1.4). Note that, the concept and tec hnique concernin g the non-zero-sum SDG and P areto optimalit y used in this pap er is reﬁned and generalize d from Dai [18] and Karatzas and Li [31 ]. The fourth aim of the pap er in v olv es three folds. First,we study some queueing netw orks (see, e.g., Figure 4) w hose dyn amics (e.g., queue length pr o cess) is go v erned by sp eciﬁc forms of the FB-SDEs in (1 .3). These forms can b e a L ´ evy driv en SDE, a p -dimensional reﬂecting Bro wnian motion (RBM) (see, e.g., Dai [14 ], Dai and Dai [12], Dai and Jiang [21]), or a reﬂecting diﬀu s ion with regime switc hing (RDRS) (see, e.g., Dai [18]). The reﬂecting diﬀusion is the functional limit of a sequence of physical queueing p ro cesses under diﬀu s iv e scaling, a general completely- S b ound ary reﬂection constrain t, and a wel l-kno wn hea vy traﬃc condition (an a nalogous treatmen t as t he one for “ inﬁn ite constan t” in the KPZ equat ion (see, e.g., Hairer [26])). In realit y , the c haracteristics of L ´ evy dr iv en net w orks ma y b e used to mo del or approximate more general batc h-arriv al and batc h-service queu eing net works. Second, w e discuss ho w to use the queueing systems and their asso ciated reﬂecting diﬀusion appro ximations to motiv at e the SDG problem. The criterion f or the pla y ers in the game 7 Figure 4: A queueing netw or k system with p -job classe s can b e the qu eue length based p erf ormance optimization ones or queueing related cost/proﬁt optimization ones. Third , we stud y the applications of the FB-SPDEs present ed by (1.1) in the qu eueing n et works. Th er e are tw o t yp es of equ ations inv olv ed. One is the Kolmogoro v’s equation or F okk er-Planc k’s form u la oriente d PDEs/SPDEs, whic h are corresp ond ing to the distributions of queueing length pro cesses und er giv en net work cont rol rules. This t yp e of equations are mainly used to estimate the p erformance measures of the qu eueing n et works. Another t yp e of equations are the HJB equation orien ted PDES/SPDES, which are m ainly used to obtain optimal con trol rules o v er the set of admissible strategies for the queueing net works. The remainder of the pap er is organized as follo w s. In Sectio n 2, we introd uce suitable functional top ological space and state conditions required for our main th eorems to guarant ee the we ll-p osedness of an adapted 4-tuple strong solution to the u niﬁed system of coupled FB- SPDEs in (1.1). In Secti on 3, w e study the uniﬁed system of coupled FB-SDEs with L ´ evy jumps and sk ew r eﬂections in (1.3) a nd presen t the well-posedn ess theorem. In p articular, w e establish the solution connection b et w een th e FB-SPDEs and the FB-SDEs. In S ection 4, w e formally formulate the FB-SDEs based SGE problem in (1.4) and determine the Pa reto optimal Nash equ ilibrium p olicy p ro cess b y a sy s tem of generalized stochastic HJ B equations (a particular form of th e coupled FB-SPDEs). Relate d applications in queueing netw orks are also d iscussed. F inally , in Sections 5-7, w e dev elop theory to pro ve our main theorems. 2 The Uniﬁed System of Coupled FB-SP DEs wit h L´ evy Jumps First o f all, let (Ω , F , P ) b e a ﬁxed complete probabilit y space. Then, we deﬁn e a stan- dard d -dimensional Bro wn ian motion W ≡ { W ( t ) , t ∈ [0 , T ] } for a giv en T ∈ [0 , ∞ ) wit h 8 W ( t ) = ( W 1 ( t ) , ..., W d ( t )) ′ and a h -dimensional general L ´ evy pure jum p pro cess (or sp ecial sub ord inator) L ≡ { L ( t ) , t ∈ [0 , T ] } with L ( t ) ≡ ( L 1 ( t ) , ..., L h ( t )) ′ on the space (see, e.g., Ap- plebaum [1], Bertoin [6 ], and Sato [48]). Note that the pr ime app eared in this p ap er is u sed to denote the corresp onding transp ose of a matrix or a v ector. F u rthermore, W , L , and their comp onent s are supp osed to b e indep end en t of eac h other. F or eac h λ = ( λ 1 , ...λ h ) ′ > 0, w hic h is called a reve rsion rate vec tor in man y applications, we let L ( λs ) = ( L 1 ( λ 1 s ) , ..., L h ( λ h s )) ′ . Then, w e d enote a ﬁltration b y {F t } t ≥ 0 with F t ≡ σ {G , W ( s ) , L ( λs ) : 0 ≤ s ≤ t } for eac h t ∈ [0 , T ], w here G is σ -algebra indep end en t of W and L . In addition, let I A ( · ) b e the index function o v er the set A and ν i for ea c h i ∈ { 1 , ..., h } b e a L ´ evy measure. T hen, we use N i ((0 , t ] × A ) ≡ P 0 0. Similarly , for eac h A ∈ {I , ¯ I } , w e sup p ose that h X i =1 Z Z    A ( c + l + o ) i ( s, x, u, v , ¯ v , ˜ v , z i )    2 λ i ν ( dz i ) (2.20) ≤ K D ,c  δ 0 c + k u k 2 C k + c ( D, r ) + k v k 2 C k + c ( D, q ) + k ¯ v k 2 C k + c ( D, q d ) + k ˜ v k 2 ν,k + c  . Then, w e can state our main theorem of this subsection as follo ws. Theorem 2.1 Supp ose that ( G, H ) ∈ L 2 G (Ω , C ∞ ( D ; R r )) × L 2 F T (Ω , C ∞ ( D ; R q )) and c ondi- tions in (2.17)-(2.20) ar e true. F urthermor e, assume that e ach A ∈ {L , ¯ L , J , ¯ J , I , ¯ I } is {F t } -adapte d for every ﬁxe d x ∈ D , z ∈ Z h , and any giv e n ( u, v , ¯ v , ˜ v ) ∈ V ∞ ( D ) with L ( · , x, 0) ∈ L 2 F ([0 , T ] , C ∞ ( D , R r )) , (2.21) J ( · , x, 0) ∈ L 2 F  [0 , T ] , C ∞ ( D , R r × d )  , (2.22) I ( · , x, 0 , · ) ∈ L 2 F  [0 , T ] × R h + , C ∞ ( D , R r × h )  , (2.23) ¯ L ( · , x, 0) ∈ L 2 F ([0 , T ] , C ∞ ( D , R q )) , (2.24) ¯ J ( · , x, 0) ∈ L 2 F  [0 , T ] , C ∞ ( D , R q × d )  , (2.25) ¯ I ( · , x, 0 , · ) ∈ L 2 F  [0 , T ] × R h + , C ∞ ( D , R q × h )  . (2.26) Then, ther e exists a unique adapte d 4-tuple str ong solution to the system in (1.1), i .e., ( U, V , ¯ V , ˜ V ) ∈ Q 2 F ([0 , T ] × D ) , (2.27) and ( U, V )( · , x ) is c` ad l` ag for e ach x ∈ D alm ost sur ely (a.s.). The pro of of Theorem 2.1 is p ro vided in S ection 5 . 2.2 T he Complex-V alued System with Op en P osition P arametric Dom ain In this subsection, we generalize the stu d y in Subsection 2.1 to the case corresp ondin g to an op en (or partially op en) p osition parametric domain D (e .g., R p or R p + ). More exac tly , 12 w e assu me that there exists a sequence of nondecreasing closed and connected sets { D n , n ∈ { 0 , 1 , ... }} su c h that D = ∞ [ n =0 D n . (2.28) F urtherm ore, let C ∞ ( D , C l ) with l ∈ { r, q } b e the Banac h space end ow ed with the norm for eac h f in th e space k f k 2 C ∞ ( D, l ) ≡ ∞ X n =0 ξ ( n + 1) k f k 2 C ∞ ( D n ,l ) , (2.29) where C l is the l -dimensional complex Euclidean space and the n orm k f k 2 C ∞ ( D n ,l ) in (2.2 9 ) is in terpreted in the corresp onding complex-v alued sense. In addition, deﬁn e ¯ Q 2 F ([0 , τ ] × D ) to b e the corresp onding sp ace in (2.11) if the terminal time T is replaced b y the stopping time τ in (1.1) an d the norm in (2.7) is s ubstituted by the one in (2.29). Finally , we use the same w a y to inte rpr et the spaces L 2 G (Ω , C ∞ ( D ; R r )) and L 2 F τ (Ω , C ∞ ( D ; R q )). Then, w e ha ve th e follo wing theorem. Theorem 2.2 Supp ose that ( G, H ) ∈ L 2 G (Ω , C ∞ ( D ; R r )) × L 2 F τ (Ω , C ∞ ( D ; R q )) and the sys- tem in (1.1) satisﬁes the c on ditions in (2.17)-(2.20) over D n for e ach n ∈ { 0 , 1 , ... } with asso ciate d (lo c al) line ar gr owth and Lipsh itz c onstant K D n ,c . F urthermor e , assume that e ach A ∈ {L , ¯ L , J , ¯ J , I , ¯ I } is {F t } -adapte d for every ﬁxe d x ∈ D , z ∈ Z h , and any g i ven ( u, v , ¯ v , ˜ v ) ∈ V ∞ ( D ) with c onditions in (2.2 1)-(2.26) b eing true. Then, the system in (1.1) has a unique adapte d 4-tuple str ong solution ( U, V , ¯ V , ˜ V ) ∈ ¯ Q 2 F ([0 , τ ] × D ) , (2.30) and ( U, V )( · , x ) is c` ad l` ag for e ach x ∈ D a.s. The pro of of Theorem 2.2 is p ro vided in S ection 5 . 2.3 I llustrativ e Examples In this subsection, w e pr esen t t w o illustr ative examples related to h eat diﬀusions and quan tum Hall/anomal ous Hall eﬀects. Mathematically , the heat diﬀusions are m o deled as a stochastic Diric hlet-Po isson problem and the Hall eﬀects are presented b y a sto chastic Schr ¨ o dinger equation. Examples relat ed to qu eueing systems an d SDGs will b e p resen ted in Section 4 after studying the system of coupled FB-SDEs in (1.3). • Heat Diﬀusions: Sto chastic Diric hlet-Po isson Pro blem 13 F or this example, we consider a sp ecial B-SPDE (a sp eciﬁc bac k ward part of the system in (1.1)). More precisely , th e asso ciated partial diﬀerenti al op erators are giv en b y ¯ L ( t, x, U, V , ¯ V , ˜ V ) ≡ − g ( t, x ) + 1 2 p X j =1 ∂ 2 V ( t, x ) ∂ x 2 j , ¯ J ( t, x, U, V , ¯ V , ˜ V ) ≡ 0 , ¯ I ( t, x, U, V , ¯ V , ˜ V , z ) ≡ 0 , where V is single-dimensional (i.e., q = 1) and g ( t, x ) is some integ rable function. Then, w e can obtain the corresp on d ing B-SPDE with jumps as follo ws, V ( t, x ) = H ( x ) + Z τ t   − g ( t, x ) + 1 2 p X j =1 ∂ 2 V ( t, x ) ∂ x 2 j   ds (2.31) − Z τ t ¯ V ( s, x ) dW ( s ) − Z τ t Z z > 0 ˜ V ( s − , x, z ) ˜ N ( λds, dz ) . When the terminal v alue H ( x ) is a b oundary based condition ov er D , we will call the r elated resolution p r oblem a sto chastic Dirichlet-Poisson pr oblem with jumps , whic h is a rand omized general form of the classical Diric hlet-P oisson problem (see, e. g., the deﬁ nition of a classic case in ∅ ksendal [39]). An explanation ab out how to use this problem to estimate the inn er or surface temp erature of certain material or an ob ject (e.g., Sun) is displa y ed in Figure 5 . Ph ysically , the randomized heat equation in (2.31 ) is deriv ed fr om a particle system jus t lik e its classic coun terpart, wh ic h can b e mod eled by a diﬀusion pro cess as follo ws, dX t = b ( X t ) dt + σ ( X t ) dW ( t ) . (2.32) No w, we use τ x D to denote the ﬁ rst exit time (a stopping time) of the pro cess X t from the time-space domain [0 , T ] × D , i.e., τ x D = inf { t > 0 , ( t, X t ) / ∈ [0 , T ] × D } , (2.33) where the upp er index x means that X t starts from x ∈ D . Then, we can imp ose a terminal- b ound ary condition with τ = τ x D as follo ws, lim t → τ x D ( ω ) V ( t, X t ) = H ( τ x D ( ω )) = H D ( x, ω ) a.s. Q x (2.34) for all ( t, x ) ∈ [0 , T ] × D . In th e case that H D ( x, ω ) is a random v ariable in dep end ent of x , the required smo oth condition for the well- p osedness of the B-SPDE in (2.31) is satisﬁed. In general, H D ( x, ω ) can b e app ro x im ated by suﬃciently smo oth function in x as required. • Hall Eﬀects: Sto c hastic Schr ¨ o dinger Equation 14 Figure 5: A heat diﬀusion system by a solution to the Diric hlet-Poisson problem In quan tum ph ysics and statistica l mec han ics, some ph enomena su c h as Hall/anomalous Hall eﬀects (see, e.g., Hall [25], Karplus and Lu ttinger [32 ], and the summ arized description at Wikip edia website) are ma jor concerns. By the deﬁnition of Hall eﬀect, the mo vemen ts of quan tu m particles (e.g., electrons) within a semiconduct/sup er cond uct are along reg ular paths (see Figure 6 for suc h an example) if the Lorentz force generated b y an external magnetic ﬁ eld w ith a p erp end icular comp onen t is imp osed. When this phenomenon h app ens, the colli sions of quantum particles will b e signiﬁcan tly redu ced and the p erformance of the semiconduct/sup ercondu ct will b e largely impro v ed. Ho w ev er, in a real ap p lication, imp osing an external magnetic ﬁeld is frequently exp ens iv e. T hus, p eople tr y to dev elop some magnetic material based semicondu ct/sup erconduct in order that the Hall eﬀect happ ens naturally (see, e.g., K arplus and Lu ttinger [32], Ch ang et al. [10]). Th is phenomenon is called anomalous Hall eﬀect. Besides observing the Hall/anomalous Hall eﬀects by physical exp erimen ts (see, e.g., Hall [25 ] and Chang et al. [1 0]), one can also analyti cally study and sim u late these eﬀects through a Sc hr ¨ o dinger equation (see, e.g., Thouless [5 1], Chai [8, 9], and the s ummarized descriptions ab out d ensit y functional theory and ti me-dep end en t d en sit y functional theory at Wikip edia website). The Sc h r ¨ o d in ger equation used in most existing studies is a form of the F okk er-Planc k’s formula (see, e.g., ∅ ksendal [39]). Here, by taking a form of L in the forw ard part of the sys tem in (1.1), we can unify these Schr ¨ o dinger equations (see, e.g., Bouard and Debussc he [4, 5], Thouless [51 ], Ch ai [8, 9]) in to the gener alize d sto cha stic nonline ar 15 Figure 6: Edg e s tates carry the cur rent Schr ¨ o dinger e quation w ith ab s orbing b oundaries for eac h i ∈ { 1 , ..., 2 p } ), idV ( t, x ) = L ( t − , x, V ) dt + J ( t − , x, V ) dW ( t ) + Z z > 0 I ( t − , x, V , z ) ˜ N ( λdt, dz ) , (2.35) where V is sin gle-dimensional (i.e., q = 1), i is the imaginary n u m b er, and L is a form of the op erator, L ( t, x, V , · ) = p X j =1 a j j ( x ) ∂ 2 V ( t, x ) ∂ x 2 j + p X j =1 b j ( x ) ∂ V ( t, x ) ∂ x j + c ( x, V ) V ( t, x ) . (2.36) Note that c ( x, V ) in (2.36) is the p oten tial, whic h m ay dep end on external temp erature and/or external magnetic ﬁeld. F or example, th e recent d isco very ab out the Anomalous Hall Eﬀect (see , e.g., Chang et al. [10]) is based on a lo wer temp erature and without imp osing external magnetic ﬁeld. F urthermore, the related Schr ¨ o dinger equation based stud ies can b e found in Ch ai [8, 9], etc. No w, if the densities app eared in the Hall/Anomalo us Hall Eﬀects are the target station- ary d istributions (i.e., the termin al v alues H ( T , x ) in (1.1) are giv en), we can tak e ¯ L in the system of (1.1) to b e a form of th e op erator in (2.36). Th en, we can ﬁn d the initial and tran- sien t distributions of quantum particles b y th e backw ard part of th e system in (1.1). F rom physic al viewp oin t, this study provides insigh ts ab out how to c haracterize and manufact ure the magnetic material based semiconductor/sup ercond uct. 16 3 The Coupled FB-SDEs with L´ evy Jumps and Sk ew Reﬂections 3.1 T he Coupled FB-SDEs and Its W ell-P osedness In this section, w e su p p ose that the pr o cess X go v erned b y th e forw ard SDE in (1.3) liv es in a state space D (e.g., a p -dimen sional p ositive orthan t or a p -dimens ional rectangle). F urtherm ore, let D i = { x ∈ R p , x · n i = b i } for i ∈ { 1 , ..., b } b e the i th b ou n dary fac e of D , where b i = 0 for i ∈ { 1 , ..., p } , b i is some p ositiv e constan t for i ∈ { p + 1 , ..., b } , and n i is the in ward un it normal v ector on the b oundary face D i . F or con venience, we deﬁne N = ( n 1 , ..., n b ). In addition, let R in (1.3) b e a p × b matrix with b ∈ { p, 2 p } , whose i th column denoted by p -dimensional ve ctor v i is the reﬂection direction of X on D i . The pro cess Y in (1.3) is a nondecreasing predictable p ro cess with Y (0 ) = 0 and boun dary regula ting prop erty as explained in (1.3). In queueing system, this pro cess is called b ou n dary idle time or b lo c kin g pro cess. Similarly , w e assume that V take s v alue in a region ¯ D with b ound ary face ¯ D i = { v ∈ R q , v · ¯ n i = ¯ b i } for i ∈ { 1 , ..., ¯ b } , where ¯ n i is the inw ard u nit normal ve ctor on th e b oundary face ¯ D i . F or con v enience, we deﬁne ¯ N = ( ¯ n 1 , ..., ¯ n ¯ b ). In ﬁnance, the give n co nstant ¯ b i is called ea rly exercise r eward. F u rthermore, S in (1.3) is sup p osed to b e a q × ¯ b matrix for a kno wn ¯ b ∈ { q , 2 q } . In addition, F ( · ) in (1.3) is a nond ecreasing predictable pro cess with F (0 ) = 0 and b oundary r egulating prop erty as exp lained in (1.3). T o guaran tee the existence and uniqueness of an adapted 6- tuple wea k solution to the coupled FB-SDEs in (1.3), we need to in tro duce the completely- S condition on the reﬂection matrix R (and similarly on S ). Deﬁnition 3.1 A p × p squar e matrix R is c al le d c ompl etely- S if and only if ther e is x > 0 such that ˜ Rx > 0 for e ach princip al sub-matrix ˜ R of R , wher e th e ve ctor ine qualities ar e to b e interpr ete d c omp onentwise. F urthermor e, a p × b matrix R is c al le d c ompletely- S if and only if e ach p × p squ ar e sub-matrix of N ′ R is c ompletely-S. Note that th e completely- S c ondition on the reﬂection matrices guaran tees th at the cou- pled FB-SDEs are of in w ard reﬂection on eac h b oundary and corner of the orthant or the rectangle (see, e.g. , Figure 2 and Dai [14 ]). F urthermore, the reﬂection ap p eared here is called ske w r eﬂection that is a generalizatio n of the con ven tional mirror (or called symmetry) reﬂection. No w, the co eﬃcien t functions giv en in (1.3) are assu med to b e {F t } -predictable and are detailed as follo ws, b ( t, x, u ) ≡ b ( t, x, v , ¯ v , ˜ v , u, · ) : [0 , T ] × R p × R q × R q × d × R q × h × U → R p , σ ( t, x, u ) ≡ σ ( t, x, v , ¯ v , ˜ v , u, · ) : [0 , T ] × R p × R q × R q × d × R q × h × U → R p × d , η ( t, x, u ) ≡ η ( t, x, v , ¯ v , ˜ v , u, z , · ) : [0 , T ] × R p × R q × R q × d × R q × h × U × Z h → R p × h , c ( t, x, u ) ≡ c ( t, x, v , ¯ v , ˜ v , u, · ) : [0 , T ] × R p × R q × R q × d × R q × h × U → R q , α ( t, x, u ) ≡ σ ( t, x, v , ¯ v , ˜ v , u, · ) : [0 , T ] × R p × R q × R q × d × R q × h × U → R q × d , ζ ( t, x, u ) ≡ γ ( t, x, v , ¯ v , ˜ v , u, z , · ) : [0 , T ] × R p × R q × R q × d × R q × h × U × Z h → R q × h . 17 F or f , f 1 , f 2 ∈ { b, σ, c, α } , we sup p ose that k f ( u ) k ≤ L ( t, ω ) ( 1 + k x k + k v k + k ¯ v k + k ˜ v k ν ) , (3.1)   f 2 ( u ) − f 1 ( u )   ≤ L ( t, ω )    x 2 − x 1   +   v 2 − v 1   (3.2) +   ¯ v 2 − ¯ v 1   +   ˜ v 2 − ˜ v 1   ν  . F urtherm ore, for eac h f , f 1 , f 2 ∈ { γ , ζ } and z ∈ Z h , w e supp ose that h X i =1 Z Z k f i ( u, z i ) k 2 λ i ν i ( dz i ) (3.3) ≤ L 2 ( t, ω )  1 + k x k 2 + k v k 2 + k ¯ v k 2 + k ˜ v k 2 ν  , where f i is th e i th column of f , and h X i =1 Z Z   f 2 i ( u, z i ) − f 1 i ( u, z i )   2 λ i ν i ( dz i ) (3.4) ≤ L 2 ( t, ω )    x 2 − x 1   2 +   v 2 − v 1   2 +   ¯ v 2 − ¯ v 1   2 +   ˜ v 2 − ˜ v 1   2 ν  . In addition, we assume that the terminal v alue H ( x ) ≡ H ( x, · ) satisﬁes th e condition, k H ( x ) k ≤ L ( t, ω )(1 + k x k ) . (3.5) Finally , L in (3.1)-(3.4) and (3.5) is assumed to b e a kno wn non-negativ e sto c hastic pro cess that is {F t } -adapted and mean-squ arely in tegrable, i.e., E  Z T 0 L 2 ( t ) dt  < ∞ . (3.6) Theorem 3.1 Under c onditions (3.1)-(3.6), the fol lowing two claims ar e true: 1. If S and R satisfy th e c ompletely- S c ondition, ther e exists a unique adapte d 6-tuple we ak solution to the system in (1.3) when at le ast one of the forwar d and b ackwar d SDEs has r eﬂe ction b oundary; 2. F urthermor e, if e ach q × q su b -princip al matrix of ¯ N ′ S and e ach p × p sub- princip al matrix of N ′ R ar e invertible or if b oth of the SDEs have no r eﬂe ction b oundaries, ther e is a unique adapte d 6-tuple str ong solution to the system in (1.3). Due to the length, the pro of of Th eorem 3.1 is p ostp oned to Section 6. 18 3.2 Resolution via Coupled FB-SPDEs In this su bsection, w e consider a particular case of the coupled FB-SPDEs in (1.1) but with an additional equ ation, w hic h corresp onds to the sp ecial forms of partial diﬀerent ial op erators ¯ L , ¯ J , and ¯ I . M ore precisely , for eac h l ∈ { 0 , 1 , ..., q } , w e deﬁne ¯ L l ( t, x, U, V , ¯ V , ˜ V , u ) (3.7) ≡ p X i,j =1 ( σ σ ′ ) ij ( t, x, u ) ∂ 2 V l ( t, x ) ∂ x i ∂ x j + p X i =1   b i ( t, x, u ) + b X j =1 v ij γ j ( t, x )   ∂ V l ( t, x ) ∂ x i + d X j =1 p X i =1 σ j i ( t, x, u ) ∂ α lj ( t, x, u ) ∂ x i − c l ( t, x, u ) + q X k =1 s lk β k ( t, x ) − h X j =1 Z Z V l ( t, x + η j ( t, x, u, z j )) − V l ( t, x ) − p X i =1 ∂ V l ( t, x ) ∂ x i η ij ( t, x, u, z j ) ! ν j ( dz j ) − h X j =1 Z Z  ˜ ζ lj ( t, x + η j ( t, x, u, z j )) , u, z j ) − ˜ ζ lj ( t, x, u, z j )  ν j ( dz j ) , where η ij and η j for i ∈ { 1 , ..., p } and j ∈ { 1 , ..., h } are the ( i, j )th en try and the j th column of η resp ectiv ely . F urthermore, c 0 ( t, x, u ) = q X l =1 c l ( t, x, u ) , (3.8) ˜ ζ 0 j ( t, x, u, z j )) = q X l =1 ζ lj ( t, x, u, z j )) , (3 .9) and ζ lj for l ∈ { 1 , ..., q } and j ∈ { 1 , ..., h } is the ( i, j )th entry of ζ . In add ition, γ j ( t, x ) for j ∈ { 1 , ..., b } and β k ( t, x ) for k ∈ { 1 , ..., q } are some fun ctions in t and x . Note that, the partial deriv ativ e ∂ α lj ( t, x, u ) ∂ x i for eac h l ∈ { 0 , 1 , ..., q } , i ∈ { 1 , ..., p } , j ∈ { 1 , ..., d } should b e inte rpreted according to c hain r u le since α ( t, x ) is also a f u nction in x through ( V , ¯ V , ˜ V )( t, x ) and u ( t, x ), wh ere α 0 j ( t, x, u ) = q X l =1 α lj ( t, x, u ) . (3.10) Finally , we deﬁne ¯ J ( t, x, U, V , ¯ V , ˜ V ) = − ¯ V ( t, x ) , (3.11) ¯ I ( t, x, U, V , ¯ V , ˜ V , z ) = − ˜ V ( t, x ) , (3.12) V ( T , x ) = H ( x ) , (3.13) where, w e assume that H ∈ L 2 F T (Ω , C ∞ ( D ; R q )). Th en, w e hav e the follo wing d eﬁnition. 19 Deﬁnition 3.2 C is c al le d the admissible set of adap te d c ontr ol pr o c esses if { ¯ L l ( t, x, U, V , ¯ V , ˜ V , u ) , l ∈ { 0 , 1 , ..., q }} to ge ther with {L , J , I , ¯ J , ¯ I } satisfy the c onditions as state d in The- or em 2.1 (or The or em 2.2). Theorem 3.2 L et ( U ( t, x ) , V ( t, x ) , ¯ V ( t, x ) , ˜ V ( t, x, · )) b e the unique adapte d 4-tuple str ong solution to the ( r , q + 1) -dimensional c ouple d FB-SPDE s in (1.1 ), which c orr esp ond s to sp e ciﬁc { ¯ L , ¯ J , ¯ I } in (3.7)-(3.12), terminal c ondition i n (3.13), and a c ontr ol pr o c ess u ∈ C . If S and R satisfy the c ompletely- S c ondition, the fol lowing claims in ar e true: 1. Ther e exists a u ni q ue adapte d 6-tuple we ak solution (( X ( t ) , Y ( t )) , ( V ( t ) , ¯ V ( t ) , ˜ V ( t, z ) , F ( t ))) to the system in (1.3) when at le ast one of the SDEs has r eﬂe ction b oundary, wher e V l ( t ) = V l ( t, X ( t )) , (3.14) ¯ V lj ( t ) = − α lj ( t, X ( t ) , u ) + p X i =1 σ li ( t, X ( t ) , u ) ∂ V l ( t, X ( t )) ∂ x i ! , (3.15) ˜ V lj ( t, z )) = − ( V l ( t, X ( t ) + η j ( t, X ( t ) , u, z j )) − V l ( t, X ( t ))) (3.16) − ζ lj ( t, X ( t ) + η j ( t, X ( t ) , u, z j ) , u, z j ) for l ∈ { 0 , 1 , ..., q } and j ∈ { 1 , ..., h } , wher e α ( t, X ( t ) , u ) = α ( t, X ( t ) , V ( t, X ( t )) , ¯ V ( t, X ( t )) , ˜ V ( t, X ( t ) , · ) , u ( t, X ( t )) , · ) , (3.17) η ( t, X ( t ) , u ) = η ( t, X ( t ) , V ( t, X ( t )) , ¯ V ( t, X ( t )) , ˜ V ( t, X ( t ) , · ) , u ( t, X ( t )) , · ) , (3.18) ζ ( t, X ( t ) , u ) = ζ ( t, X ( t ) , V ( t, X ( t )) , ¯ V ( t, X ( t )) , ˜ V ( t, X ( t ) , · ) , u ( t, X ( t )) , · ); (3.19) 2. Ther e is a unique adapte d 6-tuple str ong solution to the system in (1.3) when e ach q × q sub-princip al matrix of ¯ N ′ S and e ach p × p sub-princip al matrix of N ′ R ar e invertible or wh en b oth of the SDEs have no r eﬂe ction b oundaries. 4 Connections to Non-Zero-Sum SDGs a nd Queues 4.1 Non-Z e ro-Sum SDGs By T heorem 3.2, we sup p ose that the 4-tuple ( X, V , ¯ V , ˜ V ) in (4.1) is part of a solution ( X, Y , V , ¯ V , ˜ V , F ) to th e non-Markvian system of coupled FB-SDEs with L ´ evy jumps and sk ew reﬂections in (1.3). Then, let u ( · ) b e the corresp onding B -v alued ( B ⊂ R q ) and {F t } - adapted control pro cess, wh ose l th comp onent u l ( · ) for eac h l ∈ { 1 , ..., q } is the l th pla yer’s con trol p olicy . F urthermore, w e assume that the utilit y function for eac h pla y er l ∈ { 1 , ..., q } is d eﬁned b y ( c l ( t, X ( t ) , u ) ≡ c l ( t, X ( t ) , V ( t, X ( t )) , ¯ V ( t, X ( t )) , ˜ V ( t, X ( t )) , u ( t, X ( t ))) , c 0 ( t, X ( t ) , u ) ≡ P q l =1 c l ( t, X ( t ) , u ) , (4.1) Th us, it follo ws from (3.18)-(3.19 ), (3.9)-(3.10), and (4.1) that the v alue functions { V u l (0) , l ∈ { 1 , ..., q }} in (1.4) are now w ell deﬁned. Then, w e can in tro d uce th e follo wing concepts. 20 Deﬁnition 4.1 By a non-zer o-sum SDG to the system in (1.3), we me an that e ach pl ayer l ∈ { 1 , ..., q } cho oses an optimal p olicy to maximize his own value fu nc tion expr esse d in (1.4). F urthermor e, the v alue functions { V u l (0) , l ∈ { 1 , ..., q }} do not have to add up to a c onstant (e.g., zer o), or in other wor ds, the SDG is not ne c e ssarily a zer o-sum one. Deﬁnition 4.2 u ∗ ( · ) is c al le d a P ar eto optimal N ash e quilibriu m p olicy pr o c ess if, the pr o c ess is also an optimal one to the sum of al l the q players’ value functions at time zer o; no player wil l pr oﬁt by unilater al ly changing his own p olicy when al l the other players’ p olicies ke ep the same. Mathema tic al ly, V u ∗ 0 (0) ≥ V u 0 (0) , V u ∗ l (0) ≥ V u ∗ − l l (0) (4.2) for e ach l ∈ { 0 , 1 , ..., q } and any giv e n admissible c ontr ol p olicy u , wher e u ∗ − l = ( u ∗ 1 , ..., u ∗ l − 1 , u l , u ∗ l +1 , ..., u ∗ q ) . Deﬁnition 4.3 { ¯ L l ( t, x, U, V , ¯ V , ˜ V , u ) , l ∈ { 0 , 1 , ..., q }} to gether with {L , J , I } ar e c al le d sat- isfying the c omp arison principle in terms of u if, for any two u i ∈ C with i ∈ { 1 , 2 } and any two F T -me asur able H i with asso ciate d two solutions ( U i , V i , ¯ V i , ˜ V i )( t, x ) , r esp e ctively, of (1.3) su c h that ¯ L l ( t, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , u 1 ) ≤ ¯ L l ( t, x, U 2 , V 2 , ¯ V 2 , ˜ V 2 , u 2 ) , H 1 ( x ) ≤ H 2 ( x ) for al l ( t, x ) ∈ [0 , T ] × D , we have V 1 ( t, x ) ≤ V 2 ( t, x ) . Theorem 4.1 L et ( U ( t, x ) , V ( t, x ) , ¯ V ( t, x ) , ˜ V ( t, x, · )) b e the unique adapte d 4-tuple str ong so- lution to the ( r , q + 1) -dimensional FB-SPDEs in (1.1), which c orr esp onds to sp e ciﬁc { ¯ L , ¯ J , ¯ I } in (3.7)-(3.12), terminal c ondition in (3 .13), a nd a c ontr ol p r o c ess u ∈ C . Su pp ose that S and R satisfy the c ompletely- S c ondition. If { ¯ L l ( t, x, U, V , ¯ V , ˜ V , u ) , l ∈ { 0 , 1 , ..., q }} to gether with {L , J , I , ¯ J , ¯ I } for suitably chosen γ ( t, x ) and β ( t, x ) satisfy the c omp arison principle in terms of u , the fol lowing two claims ar e true: 1. Ther e is a Par eto optimal Nash e quilibrium p oint u ∗ ( t, X ( t )) to the non-zer o-sum SDG pr oblem in (1.4) when b oth o f the SDEs in (1.3) have no r eﬂe ction b oundaries and if γ ( t, x ) = β ( t, x ) ≡ 0 ; 2. Ther e is an appr oximat e d Par eto optimal Nash e quilibrium p oint u ∗ ( t, X ( t )) to the non- zer o-sum SDG pr oblem in (1.4) when a t le ast one of the SDEs in (1.3) ha s r eﬂe ction b oundary and if γ ( t, x ) , β ( t, x ) ar e taken to b e inﬁnitely smo oth appr oximate d fu nctions of dF dt (t,x) and d Y dt ( t, x ) in x . 21 4.2 Q ueues and Reﬂecting Diﬀusions Queueing net wo rks w id ely app ear in real-w orld applications s u c h as those in service, cloud computing, and comm u n ication systems. They t ypically consist of arriv al pro cesses, s ervice pro cesses, and b uﬀer storage s with certain kind of service reg ime and net wo rk archite cture (see, e.g., an example with p -job classes in Fig ure 4 ). T he ma jor p erformance measure for this system is the queue length pr o c ess denoted b y Q ( · ) = ( Q 1 ( · ) , ..., Q p ( · )) ′ , wh er e Q i ( t ) is the n umber of i th class jobs stored in the i th bu ﬀer f or eac h i ∈ { 1 , ..., p } at time t . Let Q (0) b e the initial queue length for th e system. Then, the qu eueing d ynamics of the sys tem can b e presen ted b y Q ( t ) = Q (0) + A ( t ) − D ( t ) , (4.3) where, th e i th co mp onent A i ( t ) of A ( t ) for eac h i ∈ { 1 , ..., p } is t he total num b er of jobs arriv ed to bu ﬀer i by time t , and the i th comp onent D i ( t ) of D ( t ) is the total num b er of jobs departed from bu ﬀer i b y time t . In the follo wing discuss ions , w e use tw o generalized w ays to c haracterize the arriv al and departure p ro cesses. First, w e assume that ea c h A i ( · ) for i ∈ { 1 , ..., p } is a time-inhomogeneous L ´ evy pro cess with in tensit y measure a i ( t, Q ( t ) , z i ) dtν i ( dz i ) that is the job arr iv al rate to buﬀer i at time t and dep ends on the queue state at time t . Similarly , we assume that eac h D i ( · ) is also a time-inhomogeneous L ´ evy pro cess with in tensity measur e d i ( t, Q ( t ) , z i ) dtν i ( dz i ) th at is the assigned service rate to b uﬀer i at time t . F urtherm ore, we assume that the routing prop ortion from buﬀer j to buﬀer i for jobs ﬁnishin g service at buﬀer j is p j i ( t, Q ( t ) , z j ). Then, by the F-SDE in (1.3) and the discussions in Applebaum [1], the queue length pro cess in (4.3) for this case can b e further expressed b y dQ i ( t ) =  Z Z a i ( t, Q ( t ) , z i ) ν i ( dz i ) (4.4) + X j 6 = i Z Z p j i ( t, Q ( t ) , z j ) d j ( t, Q ( t ) , z j ) I { Q j ( t ) > 0 } ν j ( dz j ) − Z Z d i ( t, Q ( t ) , z i ) I { Q i ( t ) > 0 } ν i ( dz i )  dt + Z Z a i ( t, Q ( t ) , z i ) ˜ N i ( dt, dz i ) + X j 6 = i Z Z p j i ( t, Q ( t ) , z j ) d j ( t, Q ( t ) , z j ) I { Q j ( t ) > 0 } ˜ N j ( dt, dz j ) − Z Z d i ( t, Q ( t ) , z i ) I { Q i ( t ) > 0 } ˜ N i ( dt, dz i ) + b X j =1 R ij ( t, Q ( t )) d Y j ( t ) , where, Z = R + , Y j ( t ) in (4.4) for eac h j ∈ { 1 , ..., b } is the Sko roho d regulator pro cess and it can increase only at time t when Q j ( t ) = 0. Note that R ( t, Q ( t )) is a reﬂection matrix th at 22 ma y b e time and queue state dep endent, and the coeﬃcients in (4.4) may b e discon tin uous at the queue state Q i ( t ) = 0. Ho we v er, since the sys tem in (4.4) is designed in a con trollable manner, the service rate d i ( s, Q ( s )) can alw a ys b e s et to b e zero w h en Q i ( t ) = 0, which implies that the reﬂection part in (4.4) can b e remo ve d. Hence, the generalized Lipsc hitz and linear gro wth conditions in (3.1)-(3.4 ) ma y b e reasonably imp osed to the system in (4.4). Th us, the system derived in (4.4) ca n b e w ell-p osed. F ur thermore, the optimal p olicies in terms of cost, proﬁt, and system p erformance can b e designed and analyzed (see, e.g., the related illustration in the co ming Su bsection 4.3 ). In terested readers can a lso ﬁnd some sp eciﬁc form ulations of the queueing system (4.4) in Mandelbaum and Ma ssey [36], Mandelbaum and P ats [37], and Konstan top oulos et al. [33], etc. Second, w e assume that b oth the arriv al and service pro cesses are describ ed b y renewa l pro cesses, renewa l rew ard p ro cesses, or d oubly sto chastic renewal pr o cesses. In this case, the driv en pro cesses f or the queueing system d o not ha v e the nice statistical prop erties suc h as memoryless and s tationary increment ones. Thus, it is usu ally imp ossible to condu ct exact analysis c oncerning t he distribution of Q ( · ). Ho w ev er, und er certain conditions (e.g., th e arriv al rates close to the asso ciated service rates), one can s h o w that the corresp on d ing se- quence of diﬀu sion-scaled queue length pro cesses con verges in distribu tion to a p -dimensional reﬂecting Bro wnian motion (RBM) (see, e.g., Dai [14 ], Dai and Dai [12], Dai and Jiang [21]), or more ge nerally , a reﬂecting diﬀusion with regime sw itc hin g (RDRS) (see, e.g., Dai [18]). In other words, we hav e that ˆ Q r ( · ) ≡ 1 r Q ( r 2 · ) ⇒ ˆ Q ( · ) along r ∈ { 1 , 2 , ... } , ( 4.5) where “ ⇒ ” means “con verges in distribution” and ˆ Q ( · ) is a RBM or a RDRS. T o b e simple, w e consider the case that the limit ˆ Q ( · ) in (4.5) is a RBM living in the state space D in tro duced in Section 3. F urthermore, let θ b e a vecto r in R p and Γ b e a p × p symmetric and p ositive deﬁnite matrix. T hen, we can in tro duce the deﬁn ition of a RBM (see, e.g, Dai [14]) as follo ws. Deﬁnition 4.4 A semimar tingale RBM asso ciate d with the data ( S , θ , Γ , R ) that has initial distribution π is a c ontinuous, {F t } -adapte d, p -dimensional pr o c ess Z deﬁne d on some ﬁlter e d pr ob ability sp ac e (Ω , F , {F t } , P ) such that under P , X ( t ) = Z ( t ) + RY ( t ) f or al l t ≥ 0 , wher e 1. X has c ontinuous p aths in S , P -a.s., 2. u nder P , Z is a p -dimensiona l Br ownian motion with drift ve ctor θ and c ovarianc e matrix Γ such that { Z ( t ) − θ t, F t , t ≥ 0 } is a martinga le and P Z − 1 (0) = π , 3. Y is a {F t } -adapte d, b -dimensio nal pr o c ess such that P -a.s., for e ach i ∈ { 1 , ..., b } , the i th c omp onent Y i of Y satisﬁes 23 (a) Y i (0) = 0 , (b) Y i is c ontinuous and non-de cr e asing, (c) Y i c an incr e ase only when Z is on the fac e D i , i.e., as given i n (1.3). F rom the physical viewp oin t o f queueing system (see, e.g., Dai [14, 18]) and the discussion in Reiman and Willia ms [47], the pu shing pro cess Y in Deﬁnition 4.4 can b e assumed to a.s. satisfy Y i ( t ) = Z t 0 I D i ( X ( s )) ds. (4.6) No w, assume that H ( x ) is the stationary distribu tion that we exp ect for th e RBM X . F or example, in realit y , it is the give n distribution of the long-run a v erage qu eue lengths among diﬀeren t users or job classes. Theoreticall y , it can b e computed by a metho d (e.g., the ﬁn ite elemen t metho d designed and imp lemen ted in Dai et al. [14, 49]). Then, we can use a B-PDE or a B-SPDE (a sp ecial form of the system in (1.1)) to get the transition f u nction at e ac h time p oint to r eac h the targeted or limiting s tationary distribution H ( x ) for the RBM X for a giv en initial distribu tion (e.g., X (0) = 0 a.s. in man y situations). Hence, the corresp ond ing p erformance measures of the ph ysical qu eueing system can b e estimated. More pr ecisely , we ha v e the follo wing theorem and related remark. Theorem 4.2 Supp ose that the r eﬂe ction matrix satisﬁes the c ompletely- S c ondition. Then, the tr ansition function of the RBM X over [0 , T ] is determine d by V ( t, x ) = H ( x ) + Z T t L ( s, x, V ) ds, (4.7) wher e V i s a 1 -dimensional function. F urthermor e, L is the fol lowing form of p art ial diﬀer- ential op er ator L ( t, x, V , · ) = ( K ( t, x, V , · ) , D 1 ( t, x, V , · ) , ..., D b ( t, x, V , · )) , (4.8) K ( t, x, V , · ) = p X i,j =1 Γ ij ∂ 2 V ( t, x ) ∂ x i ∂ x j + θ · ▽ V ( t, x ) + b X i =1 D i ( t, x, V , · ) , (4.9) D i ( t, x, V , · ) = ( v i · ▽ V ( t, x )) I D i ( x ) for x ∈ D i with i ∈ { 1 , ..., b } , (4.10) wher e ▽ V is the gr adient ve ctor of V in x and I F i is the indic ator function over the set F i . Pr oof. It follo ws fr om the completely- S condition that the RBM X is a strong Mark o v pro cess (see, e.g., Dai an d Williams [13]). Then, by app lying the It ˆ o ’s formula (see, e.g., Prot- ter [46]) and F okker-Planc k’s form ula (or called Kolmogoro v ’s forward/bac kward equations, see, e.g., ∅ ksend al [39]), w e kno w that th e claim stated in the theorem is true.  Remark 4.1 Owing to the unc ertainty err or of me asur ement, H ( x ) c ould b e r andom. F ur- thermor e , th e c o eﬃcients in (4.9) may also b e r ando m, e.g., for the c ase that the limit ˆ Q ( · ) is a RDRS. Thus, a B-SPDE c an b e intr o duc e d. F urthermo r e, the indic ator function I F i ( x ) c an b e appr oximate d by a suﬃcient smo oth function in or der to apply The or em 2.1 to the e quation in (4.7), which is r e asonable fr om the viewp oint of numeric al c omp utation. 24 4.3 Q ueueing Based Game Problem F rom the inform ation s ystem displa y ed in Figures 3-4 (presen ting a parallel-serv er queueing system w ith q = p ), we can give an explanation ab out the decision pr o cess for s uc h a game problem. In this g ame, eac h pla ye r (o r ca lled user in Dai [18]) relate s to a con trol pro cess u l ( · ) for l ∈ { 1 , ..., q } o ver certain resource p o ol (e.g., called the tr an s mission rate allo cation pro cess ov er a r andomly evo lving capacit y region in Dai [18]). In the meanwhile, eac h play er l is assigned a surrogate u tilit y fun ction c l of his submitted bid (called queue length in Dai [18 ], or the app ro x im ated queue length RBM Z in Deﬁnition 4.4) to the net w ork, the price from the net w ork to h im , and the con trol policy at eac h time p oint by the cen tral informatio n administrativ e. Then, an optimal and/or fair con trol pro cess can b e determined by the utilit y functions of al l p la yers, queueing pro cess, and the a v ailable resource constraint in a co op erativ e w ay (see, e.g., Jones [29]). 5 Pro ofs of Theorem 2.1 and T heorem 2.2 W e justify the t wo th eorems by ﬁ rst pr o vin g three lemmas in the follo wing su bsection. 5.1 T he Lemmas Lemma 5.1 Assume that the c onditions i n The or em 2.1 hold and take a qu adruplet for e ach ﬁxe d x ∈ D and z ∈ Z h , ( U 1 ( · , x ) , V 1 ( · , x ) , ¯ V 1 ( · , x ) , ˜ V 1 ( · , x, z )) ∈ Q 2 F ([0 , T ] × D ) . (5.1) Then, ther e exists another quadruplet ( U 2 ( · , x ) , V 2 ( · , x ) , ¯ V 2 ( · , x ) , ˜ V 2 ( · , x, z )) such that                                U 2 ( t, x ) = G ( x ) + R t 0 L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + R t 0 J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) dW ( s ) + R t 0 R Z h I ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) ˜ N ( λds, dz ) , V 2 ( t, x ) = H ( x ) + R T t ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + R T t  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x ) − ¯ V 2 ( s − , x )  dW ( s ) + R T t R Z h  ¯ I ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) + ˜ V 1 ( s − , x, z ) − ˜ V 2 ( s − , x, z )  ˜ N ( λds, dz ) , (5.2) wher e ( U 2 , V 2 ) is a {F t } -adapte d c` ad l` ag pr o c ess and ( ¯ V 2 , ˜ V 2 ) is the c orr esp ond ing pr e dictable pr o c ess. F urth ermor e, for e ach x ∈ D , E  Z T 0 k U 2 ( t, x ) k 2 dt  < ∞ , (5.3) E  Z T 0 k V 2 ( t, x ) k 2 dt  < ∞ , (5.4 ) 25 E  Z T 0 k ¯ V 2 ( t, x ) k 2 dt  < ∞ , (5.5 ) E " h X i =1 Z T 0 Z Z    ˜ V 2 i ( t, x, z i )    2 ν i ( dz i ) dt # < ∞ . (5.6) Pr oof. F or eac h ﬁxed x ∈ D and a quadruplet as s tated in (5.1), it follo ws from conditions (2.17)-(2.26) that L ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] , C ∞ ( D , R r )) , (5.7) J ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] , C ∞ ( D , R r × d )) , (5.8) I ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] × Z h , C ∞ ( D , R r × h )) , (5.9) ¯ L ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] , C ∞ ( D , R q )) , (5.10 ) ¯ J ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] , C ∞ ( D , R q × d )) , (5.11) ¯ I ( · , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ∈ L 2 F ([0 , T ] × Z h , C ∞ ( D , R q × h )) . (5.12) By considering L , J , a nd I in (5.7)-(5.9) as new starting L ( · , x, 0 , 0 , 0 , 0), J ( · , x , 0 , 0 , 0 , 0), and I ( · , x, 0 , 0 , 0 , 0), w e can deﬁn e U 2 b y the forw ard iteration in (5.2). F ur thermore, U 2 is a {F t } -adapted c` adl` ag pro cess that is square-in tegrable for eac h x ∈ D in the sense of (5.3). No w, consider ¯ L , ¯ J , and ¯ I in (5.10)-(5.1 2) as new starting ¯ L ( · , x, 0 , 0 , 0 , 0), ¯ J ( · , x, 0 , 0 , 0 , 0), and ¯ I ( · , x, 0 , 0 , 0 , 0). Then, it follo ws from the Martingale representati on theorem (see, e.g., Theorem 5.3.5 in page 266 of A pplebaum [1]) th at there are u nique predictable pro cesses ¯ V 2 ( · , x ) and ˜ V 2 ( · , x, z ) such that ˆ V 2 ( t, x ) (5.13) ≡ E  H ( x ) + Z T 0 ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + Z T 0  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x )  dW ( s ) + Z T 0 Z Z  ¯ I ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) + ˜ V 1 ( s − , x, z )  ˜ N ( λds, dz )     F t  = ˆ V 2 (0 , x ) + Z t 0 ¯ V 2 ( s − , x ) dW ( s ) + Z t 0 Z Z ˜ V 2 ( s − , x, z ) ˜ N ( λds, dz ) . F urtherm ore, ¯ V 2 and ˜ V 2 are square-in tegrable for eac h x ∈ D in the sense of (5.5)-(5.6), and ˆ V 2 (0 , x ) (5.14) = ˆ V 2 ( T , x ) − Z T 0 ¯ V 2 ( s − , x ) dW ( s ) − Z T 0 Z Z ˜ V 2 ( s − , x, z ) ˜ N ( λds, dz ) = H ( x ) + Z T 0 ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + Z T 0  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x ) − ¯ V 2 ( s − , x )  dW ( s ) 26 + Z T 0 Z Z  ¯ I ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) + ˜ V 1 ( s − , x, z ) − ˜ V 2 ( s − , x, z )  ˜ N ( λds, dz ) . Owing to the corollary in page 8 of Protter [46], ˆ V 2 ( · , x ) can b e tak en as a c` adl` ag pro cess. No w, deﬁ ne a pr o cess V 2 giv en by V 2 ( t, x ) = E  H ( x ) + Z T t ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds (5.15) + Z T t  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x )  dW ( s ) + Z T t Z Z  ¯ I ( s − , x, U 1 , V 1 , z ) + ˜ V 1 ( s − , x, z )  ˜ N ( λds, dz )     F t  . Th us, it follo ws from (2.19)-(2.20) and simple calc ulation that V 2 ( · , x ) is square-in tegrable in the s en se of (5.4). In addition, b y (5.13)-(5.15), w e kn ow that V 2 ( t, x ) = ˆ V 2 ( t, x ) − Z t 0 ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds (5.16) − Z t 0  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x )  dW ( s ) − Z t 0 Z Z  ¯ I ( s − , x, U 1 , V 1 , z ) + ˜ V 1 ( s − , x, z )  ˜ N ( λds, dz ) , whic h implies that V 2 ( · , x ) is a c` adl` ag pro cess. Hence, for a giv en q u adruplet in (5.1), it follo ws from (5 .13)-(5.14 ) and (5.16) that the asso ciated quadru p let ( U 2 ( · , x ), V 2 ( · , x ) , ¯ V 2 ( · , x ), ˜ V 2 ( · , x, z )) sati sﬁes the equ ation (5.2) as stated in the lemma. F urthermore, w e kno w that V 2 ( t, x ) (5.17) ≡ V 2 (0 , x ) − Z t 0 ¯ L ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 2 ) ds − Z t 0  ¯ J ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 ( s − , x ) − ¯ V 2 ( s − , x )  dW ( s ) − Z t 0 Z Z  ¯ I ( s − , x, U 1 , V 1 , ¯ V − , ˜ V 1 , z ) + ˜ V 1 ( s − , x, z ) − ˜ V 2 ( s − , x, z )  ˜ N ( λds, dz ) . Th us, w e complete the pr o of of Lemma 5.1.  Lemma 5.2 Under the c onditions of The or em 2.1, c onsider a quadruplet as in (5.1) for e ach ﬁxe d x ∈ D and z ∈ Z h . Deﬁne ( U ( t, x ) , V ( t, x ) , ¯ V ( t, x ) , ˜ V ( t, x, z )) by (5.2). Then, 27 ( U ( c ) ( · , x ) , V ( c ) ( · , x ) , ¯ V ( c ) ( · , x ) , ˜ V ( c ) ( · , x, z )) for e ach c ∈ { 0 , 1 , ..., } exists a.s. and satisﬁes                                    U ( c ) i 1 ...i p ( t, x ) = G ( c ) i 1 ...i p ( x ) + R t 0 L ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + R t 0 J ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) dW ( s ) + R t 0 R Z I ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) ˜ N ( λds, dz ) , V ( c ) i 1 ...i p ( t, x ) = H ( c ) i 1 ...i p ( x ) + R T t ¯ L ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) ds + R T t  ¯ J ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ¯ V 1 , ( c ) i 1 ...i p ( s − , x ) − ¯ V ( c ) i 1 ...i p ( s − , x )  dW ( s ) + R T t R Z  ¯ I ( c ) i 1 ...i p ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) + ˜ V 1 , ( c ) i 1 ...i p ( s − , x, z ) − ˜ V ( c ) i 1 ...i p ( s − , x, z )  ˜ N ( λds, dz ) , (5.18) wher e i 1 + ... + i p = c and i l ∈ { 0 , 1 , ..., c } with l ∈ { 1 , ..., p } . F urthermor e, ( U ( c ) i 1 ...i p , V ( c ) i 1 ...i p ) for e ach c ∈ { 0 , 1 , ... } is a {F t } -adapte d c` ad l` ag pr o c ess and ( ¯ V ( c ) i 1 ...i p , ˜ V ( c ) i 1 ...i p ) is the asso ciate d pr e dictable pr o c esses. Al l of them ar e squ ar ely-inte g r able in the senses of (5.4)-(5.6). Pr oof. Without loss of generalit y , we only consider the p oint x ∈ D , whic h is an in terior one of D . Otherwise, w e can u s e the corresp onding deriv ativ e in a one-side mann er to replace the one in the follo wing pro of. First, w e sh o w that the claim in the lemma is true f or c = 1. T o do so, for eac h giv en t ∈ [0 , T ] , x ∈ D , z ∈ Z h , and ( U 1 ( t, x ) , V 1 ( t, x ) , ¯ V 1 ( t, x ) , ˜ V 1 ( t, x, z )) as in the lemma, let ( U (1) i l ( t, x ) , V (1) i l ( t, x ) , ¯ V (1) i l ( t, x ) , ˜ V (1) i l ( t, x, z )) (5.19) b e deﬁned by (5.2) but eac h A ∈ {L , J , I , ¯ L , ¯ J , ¯ I } is r eplaced by its ﬁrst-order p artial deriv ativ e A (1) i l ∈ n L (1) i l , J (1) i l , I (1) i l , ¯ L (1) i l , ¯ J (1) i l , ¯ I (1) i l o with resp ect to x l for l ∈ { 1 , ..., p } if i l = 1. Then, w e can show that the quadru plet deﬁ ned in (5.19) for eac h l is the required ﬁrst-order partial deriv ativ e of ( U, V , ¯ V , ˜ V ) in (5.2) for the giv en ( U 1 , V 1 , ¯ V 1 , ˜ V 1 ). In fact, considering an interio r p oin t x of D , w e can tak e suﬃcient ly sm all constant δ suc h that x + δ e l ∈ D , wh ere e l is the u nit v ector wh ose l th comp onen t is one and others are zero. Without loss of g eneralit y , we assum e th at δ > 0. Th en , for eac h f ∈ { U, V , ¯ V , ˜ V , U 1 , V 1 , ¯ V 1 , ˜ V 1 } and i l = 1 with l ∈ { 1 , ..., p } , w e deﬁn e f i l ,δ ( t, x ) ≡ f ( t, x + δ e l ) . (5.20) F urtherm ore, let ∆ f (1) i l ,δ ( t, x ) = f i l ,δ ( t,x ) − f ( t,x ) δ − f (1) i l ( t, x ) , (5.21) 28 and let ∆ A (1) i l ,δ ( t, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) (5.22) = 1 δ  A ( t, x + δ e l , U 1 ( t, x + δ e l ) , V 1 ( t, x + δ e l ) , ¯ V 1 ( t, x + δ e l ) , ˜ V 1 ( t, x + δ e l , z )) −A ( t, x, U 1 ( s, x ) , V 1 ( t, x ) , ¯ V 1 ( t, x ) , ˜ V 1 ( t, x, z ))  −A (1) i l ( t, x, U 1 ( s, x ) , V 1 ( t, x ) , ¯ V 1 ( t, x ) , ˜ V 1 ( t, x, z )) for eac h A ∈ {L , J , I , ¯ L , ¯ J , ¯ I } . No w, let T r( A ) denote the trace of the matrix A ′ A for a giv en matrix A and let (T r( A )) j b e the j th term in th e su m mation of the trace. F urthermore, for eac h ﬁxed t ∈ [0 , T ], δ > 0, and γ > 0, deﬁne Z δ ( t, x ) ≡ ζ (∆ U (1) i l ,δ ( t, x ) + ∆ V (1) i l ,δ ( t, x )) (5.23) =  T r  ∆ U (1) i l ,δ ( t, x )  + T r  ∆ V (1) i l ,δ ( t, x )  e 2 γ t . Then, it follo ws from (5.17) and the It ˆ o ’s formula (see, e.g., Theorem 1.14 and Th eorem 1.16 in pages 6-9 of ∅ ksendal and S ulem [40]) that Z δ ( t, x ) + Z T t T r  ∆ ¯ J (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) (5.24) +∆ ¯ V 1 , (1) i l ,δ ( s − , x ) − ∆ ¯ V (1) i l ,δ ( s, x )  e 2 γ s ds + h X j =1 Z T t Z Z  T r  ∆ ¯ I (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) +∆ ˜ V 1 , (1) i l ,δ ( s − , x, z j ) − ∆ ˜ V (1) i l ,δ ( s − , x, z )  j e 2 γ s N j ( λ j ds, dz j ) = 2 Z t 0  − γ T r  ∆ U (1) i l ,δ ( s, x )  +  ∆ U (1) i l ,δ ( s, x )  ′  ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )   e 2 γ s ds +2 Z T t  − γ T r  ∆ V (1) i l ,δ ( s, x )  +  ∆ V (1) i l ,δ ( s, x )  ′  ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )   e 2 γ s ds − M δ ( t, x ) ≤  − 2 γ + 1 ˆ γ   Z t 0 T r  ∆ U (1) i l ,δ ( s, x )  e 2 γ s ds + Z T t T r  ∆ V (1) i l ,δ ( s, x )  e 2 γ s ds  + ˆ γ Z t 0    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds + ˆ γ Z T t    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds − M δ ( t, x ) = ˆ γ Z t 0    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds 29 + ˆ γ Z T t    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds − M δ ( t, x ) if, in th e last equalit y , we tak e ˆ γ = 1 2 γ > 0 . (5.25) Note that M δ ( t, x ) in (5.24) is a martingale of th e form, M δ ( t, x ) (5.26) = − 2 d X j =1 Z t 0  ∆ U (1) i l ,δ ( s − , x )  ′ ∆( J j ) (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) e 2 γ s dW j ( s ) − 2 h X j =1 Z t 0 Z Z  ∆ U (1) i l ,δ ( s − , x )  ′ ∆( I j ) (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z j ) e 2 γ s ˜ N j ( λ j ds, dz j ) +2 d X j =1 Z T t  ∆ V (1) i l ,δ ( s − , x )  ′  ∆( ¯ J j ) (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) +∆( ¯ V 1 j ) (1) i l ,δ ( s − , x ) − ∆( ¯ V j ) (1) i l ,δ ( s − , x )  e 2 γ s dW j ( s ) +2 h X j =1 Z T t Z Z  ∆ V (1) i l ,δ ( s − , x )  ′  ∆( ¯ I j ) (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z j ) +∆( ˜ V 1 j ) (1) i l ,δ ( s − , x, z j ) − ∆( ˜ V j ) (1) i l ,δ ( s − , x, z j )  e 2 γ s ˜ N j ( λ j ds, dz j ) . Th us, b y the martingale pr op ert y and (5.24), we kno w that E  Z δ ( t, x ) + Z T t T r  ∆ ¯ J (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) (5.27) +∆ ¯ V 1 , (1) i l ,δ ( s − , x ) − ∆ ¯ V (1) i l ,δ ( s, x )  e 2 γ s ds + h X j =1 Z T t Z Z  T r  ∆ ¯ I (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) +∆ ˜ V 1 , (1) i l ,δ ( s − , x, z ) − ∆ ˜ V (1) i l ,δ ( s − , x, z )  j e 2 γ s N j ( λ j ds, dz j )  ≤ ˆ γ E  Z t 0    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  + ˆ γ  Z T t    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  . F urtherm ore, by (5.24)-(5.2 7 ) and the Burkholder-Da vis-Gundy ’s inequalit y (see, e.g., The- orem 48 in page 193 of Protter [46]), w e ha ve the follo wing observ ation, E " sup 0 ≤ t ≤ T | M δ ( t, x ) | # (5.28) 30 ≤ ˆ γ K 1 E  Z t 0    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  + ˆ γ K 1  Z T t    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  , where K 1 is some nonnegat iv e constant depend ing only on K D , 0 , K D , 1 , T , and d . Note that, the detaile d estimatio n pro cedure for the quan tit y o n the righ t-hand side of (5.28) is p ostp oned to the same argument us ed for (5.55) in th e pro of of L emma 5.3 since m ore exact calculatio ns are requir ed there. Next, for ea c h ﬁxed t ∈ [0 , T ], x ∈ D , and σ > 0, consider the random v ariable set { Z δ ( t, x ), δ ∈ [0 , σ ] } . It follo ws f r om Lemma 1.3 in pages 6-7 of P eskir and Sh iry aev [45] that there is a counta ble subset C = { δ 1 , δ 2 , ... } ⊂ [0 , σ ] su ch that esssup δ ∈ [ 0 ,σ ] Z δ ( t, x ) = su p δ ∈C Z δ ( t, x ) , a.s. , (5.29) where “esssup” d enotes th e essentia l suprem um. F u rthermore, tak e ( ¯ Z δ 1 ( t, x ) = Z δ 1 ( t, x ) , ¯ Z δ n +1 ( t, x ) = ¯ Z δ n ( t, x ) ∨ Z δ n +1 ( t, x ) f or n ∈ { 1 , 2 , ... } , (5.30) where α ∨ β = max { α, β } for any t wo real num b ers α and β . Obvi ously , ( Z δ ( t, x ) ≤ ¯ Z δ ( t, x ) for eac h δ ∈ C ¯ Z δ 1 ( t, x ) ≤ ¯ Z δ 2 ( t, x ) for any δ 1 , δ 2 ∈ C satisfying δ 1 ≤ δ 2 . (5.31) The second inequalit y in (5.3 1 ) implies that the set  ¯ Z δ ( t, x ) , δ ∈ C  is upw ard s directed. Hence, for eac h t ∈ [0 , T ], x ∈ D , σ > 0, and the asso ciated sequence of { δ n , n = 1 , 2 , ... } , it follo ws fr om (5.29) that E  esssup 0 ≤ δ ≤ σ Z δ ( t, x )  (5.32) ≤ E  esssup δ ∈C ¯ Z δ ( t, x )  = lim n →∞ E  ¯ Z δ n ( t, x )  = lim n →∞ E  max δ ∈{ δ 1 ,...,δ n } Z δ ( t, x )  . In addition, for eac h ﬁxed n ∈ { 2 , 3 , ... } , let ¯ M δ n ( t, x ) = M δ n ( t, x ) I { Z δ n ≥ ¯ Z δ n − 1 } + M δ n − 1 ( t, x ) I { Z δ n < ¯ Z δ n − 1 } . (5.33) Th us, b y the ind uction metho d in terms of n ∈ { 1 , 2 , ... } an d (5.24), we know that E  max δ ∈{ δ 1 ,...,δ n } Z δ ( t, x )  (5.34) ≤ ˆ γ lim n →∞ E  Z t 0 max δ ∈{ δ 1 ,...,δ n }    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds 31 + Z T t max δ ∈{ δ 1 ,...,δ n }    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  − lim n →∞ E  ¯ M δ n ( t, x )  ≤ K E  Z t 0 esssup 0 ≤ δ ≤ σ    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds + Z T t esssup 0 ≤ δ ≤ σ    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds  + Z T 0 esssup 0 ≤ δ ≤ σ    ∆ J (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds + Z T 0 h X i =1 Z Z esssup 0 ≤ δ ≤ σ    ∆ I (1) i,i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z i )    2 e 2 γ s λ i ν i ( dz i ) ds # , where K is a nonnegativ e constan t dep ending only on K D , 0 , d , T , and γ . Note that, in the second in equalit y , w e ha ve u s ed th e fact in (5.27) and th e follo wing observ ation   E  ¯ M δ n ( t, x )    ≤ E h sup t ∈ [0 ,T ] k M δ n ( t, x ) k i + E h sup t ∈ [0 ,T ]   M δ n − 1 ( t, x )   i . (5.3 5) No w, recall the condition that ( U 1 ( · , x ) , V 1 ( · , x ) , ¯ V 1 ( · , x ) , ˜ V 1 ( · , x, z )) ∈ Q 2 F ([0 , T ] × D ) . Then, for eac h x ∈ D , z ∈ Z h , any c ∈ { 0 , 1 , ... } , a nd any small num b er ξ suc h that x + ξ e l ∈ D , we hav e that    ( U 1 , ( c ) ( t, x + ξ e l ) , V 1 , ( c ) ( t, x + ξ e l ) , ¯ V 1 , ( c ) ( t, x + ξ e l ) , ˜ V 1 , ( c ) ( t, x + ξ e l , z ))    (5.36) ≤      max x ∈ D    U 1 , ( c ) ( t, x )    , max x ∈ D    V 1 , ( c ) ( t, x )    , max x ∈ D    ¯ V 1 , ( c ) ( t, x )    , max x ∈ D    ˜ V 1 , ( c ) ( t, x, z )         . Note that the related quantitie s on the right-hand side of (5.36) are squarely in tegrable a.s. in term of the Leb esgue measure and/or the L ´ evy measure. Therefore, ˜ V 1 ( t, x, · ) (the integ ration of ˜ V 1 ( t, x, z ) with resp ect to the L´ evy measure) is also inﬁnitely smo oth in eac h x ∈ D due to the Leb esgue’s dominated conv ergence theorem. Th u s, by the mean-v alue theorem, there exist some constants ξ 1 ∈ (0 , δ ) and ξ ∈ (0 , ξ 1 ), whic h dep end on δ , such that ∆ A (1) i l ,δ ( t, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) (5.37) = ξ 1 A (2) i l ( t, x + ξ e l , U 1 ( t, x + ξ e l ) , V 1 ( t, x + ξ e l ) , ¯ V 1 ( t, x + ξ e l ) , ˜ V 1 ( t, x + ξ e l , · )) a.s. for eac h A ∈ {L , J , ¯ L} . Due to (5.37), (2.17), and (5.3 6), the qu an tit y on the left-hand side of (5.37) for all δ is dominated by a squ arely-in tegrable random v ariable in term s of the pro du ct measure dt × dP . Similarly , for A = ¯ J and eac h z ∈ Z h , w e a.s. hav e that ∆ A (1) i l ,δ ( t, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z ) (5.38) = ξ 1 A (2) i l ( t, x + ξ e l , U 1 ( t, x + ξ e l ) , V 1 ( t, x + ξ e l ) , ¯ V 1 ( t, x + ξ e l ) , ˜ V 1 ( t, x + ξ e l , z ) , z ) . 32 Owing to (5.37), (2.18), and (5.36), the quan tit y on t he left-hand side of (5.38) for a ll δ is dominated b y a squ arely-in tegrable random v ariable in terms of the p ro duct measure dt × ν ( dz ) × dP . Therefore, it follo ws from (5.32)-(5.34) and the Leb esgue’s d ominated con vergence theorem that lim σ → 0 E  esssup 0 ≤ δ ≤ σ Z δ ( t, x )  (5.39) ≤ K E  Z t 0 lim σ → 0 esssup 0 ≤ δ ≤ σ    ∆ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds + Z T t lim σ → 0 esssup 0 ≤ δ ≤ σ    ∆ ¯ L (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    2 e 2 γ s ds + Z T 0 lim σ → 0 esssup 0 ≤ δ ≤ σ    ∆ J (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 )    e 2 γ s ds + Z T 0 h X i =1 Z Z lim σ → 0 esssup 0 ≤ δ ≤ σ    ∆ I (1) i l ,δ ( s − , x, U 1 , V 1 , ¯ V 1 , ˜ V 1 , z i )    e 2 γ s λ i ν i ( dz i ) ds # . Hence, by (5.39) and the F atou’s lemma, we kno w that, for any sequence σ n satisfying σ n → 0 along n ∈ N , there is a su bsequence N ′ ⊂ N such that esssup 0 ≤ δ ≤ σ n Z δ ( t, x )) → 0 along n ∈ N ′ a.s. (5.40) The conv ergence in (5.40) implies that the ﬁrst-order deriv ativ es of U and V in terms of x l for eac h l ∈ { 1 , ..., p } exists. More exac tly , they equal U (1) i l ( t, x ) and V (1) i l ( t, x ) a.s. resp ectiv ely for eac h t ∈ [0 , T ] and x ∈ D . F urthermore, they are {F t } -adapted. No w, we pr o ve the claim for ¯ V . In fact, it follo ws from the pro of as in (5.32)-(5.34) that lim σ → 0 E  Z T t esssup 0 ≤ δ ≤ σ T r  ∆ ¯ J (1) i l ,δ ( s, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) (5.41) +∆( ¯ V 1 ) (1) i l ,δ ( s, x ) − ∆ ¯ V (1) i l ,δ ( s, x )  e 2 γ s ds i is also b ounded b y the quan tity on the r igh t-han d side of (5.39). Th us, b y (5.40 ) and (5.41), w e kno w that lim δ → 0 ∆ ¯ V (1) i l ,δ ( t, x ) = lim δ → 0  ∆ ¯ J (1) i l ,δ ( t, x, U 1 , V 1 , ¯ V 1 , ˜ V 1 ) + ∆( ¯ V 1 ) (1) i l ,δ ( t, x )  = 0 , a.s. Hence, the ﬁ rst-order deriv ativ e of ¯ V in x l for eac h l ∈ { 1 , ..., p } exists and equals ¯ V (1) i l ( t, x ) a.s. for every t ∈ [0 , T ] and x ∈ D . F urthermore, it is a {F t } -predictable pro cess. Similarly , w e can get the conclusion for ˜ V (1) i l ( t, x, z ) asso ciated with eac h l , t , x , and z . Second, we supp ose that ( U ( c − 1) ( t, x ) , V ( c − 1) ( t, x ) , ¯ V ( c − 1) ( t, x ) , ˜ V ( c − 1) ( t, x, z )) corresp ond- ing to a giv en ( U 1 ( t, x ) , V 1 ( t, x ), ¯ V 1 ( t, x ) , ˜ V 1 ( t, x, z )) ∈ Q 2 F ([0 , T ] × D ) exists f or any giv en c ∈ { 1 , 2 , ... } . Then, w e can show that  U ( c ) ( t, x ) , V ( c ) ( t, x ) , ¯ V ( c ) ( t, x ) , ˜ V ( c ) ( t, x, z )  (5.42) 33 exists for the giv en c ∈ { 1 , 2 , ... } . In fact, consider any ﬁxed nonnegativ e in teger num b ers i 1 , ..., i p satisfying i 1 + ... + i p = c − 1 for the giv en c ∈ { 1 , 2 , ... } . T ak e f ∈ { U, V , ¯ V , ˜ V } , l ∈ { 1 , ..., p } , and suﬃcient ly small δ > 0. Then, let f ( c − 1) i 1 ... ( i l +1) ...i p ,δ ( t, x ) ≡ f ( c − 1) i 1 ...i p ( t, x + δ e l ) (5.43) corresp ond to the ( c − 1)th-order partial deriv ativ e A ( c − 1) i 1 ...i p ( s, x + δ e l , U 1 ( s, x + δ e l ) , V 1 ( s, x + δ e l )) of A ∈ {L , J , I , ¯ L , ¯ J , ¯ I } via (5.2 ). S imilarly , let ( U ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , V ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , ¯ V ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , ˜ V ( c ) i 1 ... ( i l +1) ...i p ( t, x, z )) b e deﬁned b y (5.2), w here A ∈ {L , J , I , ¯ L , ¯ J , ¯ I } are replaced by their c th-order p artial deriv ativ es A ( c ) i 1 ... ( i l +1) ...i p corresp ondin g to a giv en t, x , U 1 ( t, x ), V 1 ( t, x ), ¯ V 1 ( t, x ), ˜ V 1 ( t, x, z ). F urtherm ore, let ∆ f ( c ) i 1 ... ( i l +1) ...i p ,δ ( t, x ) = f ( c − 1) i 1 ... ( i l +1) ...i p ,δ ( t, x ) − f ( c − 1) i 1 ...i p ( t, x ) δ − f ( c ) i 1 ... ( i l +1) ...i p ( t, x ) (5.44) for eac h f ∈ { U, V , ¯ V , ˜ V , U 1 , V 1 , ¯ V 1 , ˜ V 1 } . T h en, deﬁne ∆ A ( c ) i 1 ... ( i l +1) ...i p ,δ ( t, x, U 1 , V 1 ) (5.45) ≡ 1 δ  A ( c − 1) i 1 ...i p ( t, x + δ e l , U 1 ( t, x + δ e l ) − V 1 ( t, x + δ e l ) , · ) −A ( c − 1) i 1 ...i p ( s, x, U 1 ( s, x ) , V 1 ( s, x ) · )  −A ( c ) i 1 ... ( i l +1) ...i p ( s, x, U 1 ( s, x ) , V 1 ( s, x ) · ) for eac h A ∈ {L , J , I , ¯ L , ¯ J , ¯ I } . Thus, by the It ˆ o ’s formula and rep eating the p ro cedure as used in th e ﬁr st step, we kno w that ( U ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , V ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , ¯ V ( c ) i 1 ... ( i l +1) ...i p ( t, x ) , ˜ V ( c ) i 1 ... ( i l +1) ...i p ( t, x, z )) exist for the giv en c ∈ { 1 , 2 , ... } and all l ∈ { 1 , ..., p } . Therefore, the claim in (5.42) is true. Third, b y the indu ction m etho d w ith resp ect to c ∈ { 1 , 2 , ... } and the con tin uit y of all partial deriv ativ es in terms of x ∈ D , w e kn o w that the claims in the lemma are true. H ence, w e ﬁnish the pro of of Lemma 5.2.  T o state and pro ve the n ext lemma, let D 2 F ([0 , T ] , C ∞ ( D , R l )) with l ∈ { r , q } b e the set of R l -v alued {F t } -adapted and squarely integrable c` adl` ag p ro cesses as in (2.8). F urth ermore, for an y giv en num b er sequence γ = { γ c , c = 0 , 1 , 2 , ... } with γ c ∈ R , deﬁn e M D γ [0 , T ] to b e the follo wing Banac h space (see , e.g., the related exp lanation in Y ong and Zhou [52], and Situ [50]) M D γ [0 , T ] ≡ D 2 F ([0 , T ] , C ∞ ( D , R r )) (5.46) × D 2 F ([0 , T ] , C ∞ ( D , R q )) × L 2 F ,p ([0 , T ] , C ∞ ( D , R q × d )) × L 2 p ([0 , T ] × R h + , C ∞ ( D , R q × h )) , 34 whic h is endo wed with the norm    ( U, V , ¯ V , ˜ V )    2 M D γ ≡ ∞ X c =0 ξ ( c )    ( U, V , ¯ V , ˜ V )    2 M D γ c ,c (5.47) for an y giv en ( U, V , ¯ V , ˜ V ) ∈ M D γ [0 , T ], and    ( U, V , ¯ V , ˜ V )    2 M D γ c ,c = E " sup 0 ≤ t ≤ T k U ( t ) k 2 C c ( D, q ) e 2 γ c t # (5.48) + E " sup 0 ≤ t ≤ T k V ( t ) k 2 C c ( D, q ) e 2 γ c t # + E  Z T 0   ¯ V ( t )   2 C c ( D, q d ) e 2 γ c t dt  + E  Z T 0    ˜ V ( t )    2 ν,c e 2 γ c t dt  . Then, w e hav e the follo w ing lemma. Lemma 5.3 Under the c onditions of The or em 2.1, al l the claims in the the or em ar e true. Pr oof. By (5.2), w e can deﬁn e the follo wing map Ξ : ( U 1 ( · , x ) , V 1 ( · , x ) , ¯ V 1 ( · , x ) , ˜ V 1 ( · , x, z )) → ( U ( · , x ) , V ( · , x ) , ¯ V ( · , x ) , ˜ V ( · , x , z )) . Then, w e sh o w that Ξ forms a cont raction m apping in M D γ [0 , T ]. In fact, consider ( U i ( · , x ) , V i ( · , x ) , ¯ V i ( · , x ) , ˜ V i ( · , x, z )) ∈ M D γ [0 , T ] for eac h i ∈ { 1 , 2 , ... } , satisfying ( U i +1 ( · , x ) , V i +1 ( · , x ) , ¯ V i +1 ( · , x ) , ˜ V i +1 ( · , x, z )) = Ξ( U i ( · , x ) , V i ( · , x ) , ¯ V i ( · , x ) , ˜ V i ( · , x, z )) . F urtherm ore, deﬁn e ∆ f i = f i +1 − f i with f ∈ { U, V , ¯ V , ˜ V } and tak e ζ (∆ U i ( t, x ) + ∆ V i ( t, x )) =  T r  ∆ U i ( t, x )  + T r  ∆ V i ( t, x )  e 2 γ 0 t . (5.49) Th us, it follo ws fr om (2.1 7 ) and the similar argum ent as used in pr o vin g (5.24 ) t hat, for a γ 0 > 0 and eac h i ∈ { 2 , 3 , ... } , ζ (∆ U i ( t, x ) + ∆ V i ( t, x )) (5.50) 35 + Z T t T r  ∆ ¯ J ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +∆ ¯ V i − 1 ( s, x ) − ∆ ¯ V i ( s, x )  e 2 γ 0 s ds + h X j =1 Z T t Z Z  T r  ∆ ¯ I ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z ) +∆ ˜ V i − 1 ( s − , x, z ) − ∆ ˜ V i ( s − , x, z )  j e 2 γ 0 s N j ( λ j ds, dz j ) ≤ ˆ γ 0  Z t 0    ∆ L ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 )    2 e 2 γ 0 s ds + Z T t    ∆ ¯ L ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 )    2 e 2 γ 0 s ds  − M i ( t, x ) ≤ ˆ γ 0 K a, 0 N i − 1 ( t ) − M i ( t, x ) , where K a, 0 is some nonnegativ e constant dep ending only on K D , 0 . F or the last inequalit y in (5.50), w e hav e tak en ˆ γ 0 = 1 2 γ 0 > 0 . (5.51) F urtherm ore, N i − 1 ( t ) app eared in (5.50) is giv en by N i − 1 ( t ) (5.52) = Z t 0   ∆ U i − 1 ( s )   2 C k ( D, r ) e 2 γ 0 s ds + Z T t    ∆ V i − 1 ( s )   2 C k ( D, q ) +   ∆ ¯ V i − 1 ( s )   2 C k ( D, q d ) +    ∆ ˜ V i − 1 ( s )    2 ν,k  e 2 γ 0 s ds. In addition, M i ( t, x ) in (5.50) is a martingale of th e form, M i ( t, x ) = (5.53) − 2 d X j =1 Z t 0  ∆ U i ( s − , x )  ′ ∆ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) e 2 γ s dW j ( s ) − 2 h X j =1 Z t 0 Z Z  ∆ U i ( s − , x )  ′ ∆ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) e 2 γ s ˜ N j ( λ j ds, dz j ) +2 d X j =1 Z T t  ∆ V i ( s − , x )  ′  ∆ ¯ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +(∆ ¯ V i − 1 ) j ( s − , x ) − (∆ ¯ V i ) j ( s − , x )  e 2 γ 0 s dW j ( s ) 36 +2 h X j =1 Z T t Z Z  (∆ V i ) j ( s − , x )  ′  ∆ ¯ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) +(∆ ˜ V i − 1 ) j ( s − , x, z j ) − (∆ ˜ V i ) j ( s − , x, z j )  e 2 γ 0 s ˜ N j ( λ j ds, dz j ) . Then, it f ollo ws from (5.50)-(5.53 ) and the martingale prop erties r elated to the It ˆ o ’s sto c hastic in tegral that E  ζ (∆ U i ( t, x ) + ∆ V i ( t, x ))  e 2 γ 0 t (5.54) + Z T t T r  ∆ ¯ J ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +∆ ¯ V i − 1 ( s, x ) − ∆ ¯ V i ( s, x )  e 2 γ 0 s ds + h X j =1 Z T t Z Z  T r  ∆ I ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z ) +∆ ˜ V i − 1 ( s − , x, z ) − ∆ ˜ V i ( s − , x, z )  j e 2 γ 0 s λ j dsν j ( dz j )  ≤ ˆ γ 0 ( T + 1) K a, 0    (∆ U i − 1 , V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k . Next, it follo ws fr om (5.53) that E " sup 0 ≤ t ≤ T   M i ( t, x )   # (5.55) ≤ 2 d X j =1 E " sup 0 ≤ t ≤ T     Z t 0  ∆ U i ( s − , x )  ′ ∆ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) e 2 γ 0 s dW j ( s )    i +2 h X j =1 E " sup 0 ≤ t ≤ T     Z t 0 Z Z  ∆ U i ( s − , x )  ′ ∆ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) e 2 γ 0 s ˜ N ( λ j ds, dz j )    i +4 d X j =1 E " sup 0 ≤ t ≤ T     Z t 0  ∆ V i ( s − , x )  ′  ∆ ¯ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +(∆ ¯ V i − 1 ) j ( s − , x ) − (∆ ¯ V i ) j ( s − , x )  e 2 γ 0 s dW j ( s )    +4 h X j =1 E " sup 0 ≤ t ≤ T     Z t 0 Z Z  ∆ V i ( s − , x )  ′  ∆ ¯ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) +(∆ ˜ V i − 1 ) j ( s − , x, z j ) − (∆ ˜ V i ) j ( s − , x, z j )  e 2 γ 0 s ˜ N ( λ j ds, dz j )    i . 37 By the Burkholder-Da vis-Gund y’s inequalit y (see, e.g., T heorem 48 in page 193 of Prot- ter [46]), the right-hand side of the in equ alit y in (5.55) is b ound ed b y K b, 0   d X j =1 E  Z T 0   ∆ U i ( s − , x )   2 (5.56)    (∆ J i ) j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 )    2 e 4 γ 0 s ds  1 2 # + h X j =1 E  Z T 0 Z Z   ∆ U i ( s − , x )   2    ∆ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j )    2 e 4 γ 0 s λ j ν j ( dz j ) ds  1 2 # + d X j =1 E  Z T 0   ∆ V i ( s − , x )   2    (∆ ¯ J i ) j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +(∆ ¯ V i − 1 ) j ( s − , x ) − (∆ ¯ V i ) j ( s − , x )   2 e 4 γ 0 s ds  1 2  + h X j =1 E  Z T 0 Z Z   ∆ V i ( s − , x )   2    ∆ ¯ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) +(∆ ˜ V i ) j ( s − , x, z j ) − (∆ ˜ V i ) j ( s − , x, z j )    2 e 4 γ 0 s λ j ν j ( dz j ) ds  1 2 #! , where K b, 0 is some nonnegativ e constan t dep ending only on K D , 0 and T . F urthermore, it follo ws fr om the dir ect observ ation that the qu an tity in (5.56) is b ound ed b y K b, 0   E   sup 0 ≤ t ≤ T k ∆ U i ( t, x ) k 2 e 2 γ 0 t ! 1 2 (5.57)   d X j =1  Z T 0    ∆ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 )    2 e 2 γ 0 s ds  1 2 + h X j =1  Z T 0 Z Z    ∆ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j )    2 e 2 γ 0 s λ j ν j ( dz j ) ds  1 2 i +   E   sup 0 ≤ t ≤ T k ∆ V i ( t, x ) k 2 e 2 γ 0 t ! 1 2 38   d X j =1  Z T 0    ∆ ¯ J j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +(∆ ¯ V i − 1 ) j ( s − , x ) − (∆ ¯ V i ) j ( s − , x )   2 e 2 γ 0 s ds  1 2 + h X j =1  Z T 0 Z Z    ∆ ¯ I j ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) +(∆ ˜ V i − 1 ) j ( s − , x, z j ) − (∆ ˜ V i ) j ( s − , x, z j )    2 e 2 γ 0 s λ j ν j ( dz j ) ds  1 2 !#! . In addition, by the direct computation, we k n o w that the quant it y in (5.5 7 ) is dominated by 1 2 E " sup 0 ≤ t ≤ T k ∆ U i ( t, x ) k 2 e 2 γ 0 t # (5.58) + dK 2 b, 0 E  Z T 0 T r  ∆ J ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1  e 2 γ 0 s ds  + K 2 b, 0 E   h X j =1 Z T 0 Z Z T r  ∆ I ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j )  j e 2 γ 0 s λ j ν j ( dz j ) ds  + 1 2 E " sup 0 ≤ t ≤ T k ∆ V i ( t, x ) k 2 e 2 γ 0 t # + dK 2 b, 0 E  Z T 0 T r  ∆ ¯ J ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +∆ ¯ V i − 1 ( s − , x ) − ∆ ¯ V i ( s − , x )  e 2 γ 0 s ds  + K 2 b, 0 E   h X j =1 Z T 0 Z Z T r  ∆ ¯ I ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z j ) +∆ ˜ V i − 1 ( s − , x, z ) − ∆ ˜ V i ( s − , x, z j )  e 2 γ 0 s λ j ν j ( dz j ) ds i . Due to (5.54), the quan tit y in (5.58) is b ounded by 1 2 E " sup 0 ≤ t ≤ T k ∆ U i ( t ) k 2 C 0 ( r ) e 2 γ 0 t # + E " sup 0 ≤ t ≤ T k ∆ U i ( t ) k 2 C 0 ( q ) e 2 γ 0 t #! (5.59) + ˆ γ 0 ( T + 1) dK a, 0 K 2 b, 0    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k , where K a, 0 is some nonnegativ e constant d ep ending only on T , d , and K D , 0 . Thus, it follo w s from (2.17) and (5.50)-(5.59) that E " sup 0 ≤ t ≤ T   ∆ U i ( t )   2 C 0 ( q ) e 2 γ 0 t # + E " sup 0 ≤ t ≤ T   ∆ V i ( t )   2 C 0 ( q ) e 2 γ 0 t # (5.60) 39 ≤ 2  1 + dK 2 b, 0  K a, 0 ˆ γ 0 ( T + 1)    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k . F urtherm ore, it follo ws from (5.50) and (2.17) th at, f or i ∈ { 3 , 4 , ... } , E  Z T t T r  ∆ ¯ V i ( s, x )  e 2 γ 0 s ds  (5.61) ≤ 2 E  Z T t T r  ∆ ¯ J ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +∆ ¯ V i − 1 ( s − , x ) − ∆ ¯ V i ( s, x )  e 2 γ 0 s ds  +2 E  Z T t T r  ∆ ¯ J ( s − , x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 ) +∆ ¯ V i − 1 ( s − , x )  e 2 γ 0 s ds  ≤ 2 ˆ γ 0 K C, 0    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ 0 ,k ! , where K C, 0 is some nonn egativ e constan t dep ending only on K D , 0 and T . Similarly , it f ollo ws from (2.18) that E   h X j =1 Z T t Z Z  T r  ∆ ˜ V i ( s − , x, z )  j e 2 γ 0 s λ j dsν j ( dz j )   (5.62) ≤ 2 E   h X j =1 Z T t Z Z  T r  ∆ ¯ I ( s, x, U i , V i , ¯ V i , ¯ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z ) +∆ ˜ V i − 1 ( s − , x, z ) − ∆ ˜ V i ( s − , x, z )  j e 2 γ 0 s λ j dsν j ( dz j )  +2 E   h X j =1 Z T t Z Z  T r  ∆ ¯ I ( s, x, U i , V i , ¯ V i , ˜ V i , U i − 1 , V i − 1 , ¯ V i − 1 , ˜ V i − 1 , z ) +∆ ˜ V i − 1 ( s − , x, z )  j e 2 γ 0 s λ j dsν j ( dz j )  ≤ 2 ˆ γ 0 K C, 0    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ 0 ,k ! . Th us, b y (5. 50), (5.60)-(5.62), and the fact that all fu nctions an d norms used in this pap er are con tinuous in terms of x , we h av e    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    2 M D γ 0 , 0 (5.63) 40 ≤ ˆ γ 0 K d, 0    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ 0 ,k +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ 0 ,k ! , where K d, 0 is some nonn egativ e constan t dep ending only on K D , 0 and T . No w, by Lemma 5.2 and the similar construction as in (5.49 ), for eac h c ∈ { 1 , 2 , ... } , w e can deﬁne ζ (∆ U c,i ( t, x ) + ∆ V c,i ( t, x )) ≡  T r  ∆ U c,i ( t, x )  + T r  ∆ V c,i ( t, x )  e 2 γ c t , (5.64) where ∆ U c,i ( t, x )) = (∆ U (0) ,i ( t, x )) , ∆ U (1) ,i ( t, x )) , ..., ∆ U ( c ) ,i ( t, x )) ′ , ∆ V c,i ( t, x )) = (∆ V (0) ,i ( t, x )) , ∆ V (1) ,i ( t, x )) , ..., ∆ V ( c ) ,i ( t, x )) ′ . Then, it follo ws from the It ˆ o ’s form ula and the similar d iscu ssion f or (5.63) that    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    2 M D γ c ,c (5.65) ≤ ˆ γ c K d,c     (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ c ,k + c +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ c ,k + c  ≤ δ (( c + 1) 10 ( c + 2) 10 ... ( c + k ) 10 )( η ( c + 1) η ( c + 2) ...η ( c + k ))    (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ k + c ,k + c +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ k + c ,k + c ! , where, for the last inequalit y of (5.65), w e ha ve tak en the num b er sequence γ suc h that γ 0 < γ 1 < ... and ˆ γ c K d,c (( c + 1) 10 ( c + 2) 10 ... ( c + k ) 10 )( η ( c + 1) η ( c + 2) ...η ( c + k )) ≤ δ for some δ > 0 su c h that 2 √ e k δ is su ﬃcien tly small. Hence, w e ha v e    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    2 M D γ (5.66) ≤ e k δ     (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ  . 41 Since ( a 2 + b 2 ) 1 / 2 ≤ a + b for a, b ≥ 0, w e ha ve    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    M D γ (5.67) ≤ √ e k δ     (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    M D γ +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    M D γ  . Therefore, b y (5.67), we kno w that ∞ X i =3    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    M D γ (5.68) ≤ √ e k δ 1 − 2 √ e k δ  2    (∆ U 2 , ∆ V 2 , ∆ ¯ V 2 , ∆ ˜ V 2 )    M D γ +    (∆ U 1 , ∆ V 1 , ∆ ¯ V 1 , ∆ ˜ V 1 )    M D γ  < ∞ . Th us, from (5.68), w e s ee that ( U i , V i , ¯ V i , ˜ V i ) with i ∈ { 1 , 2 , ... } forms a Cauc hy sequence in M D γ [0 , T ], whic h implies that ther e is some ( U, V , ¯ V , ˜ V ) suc h that ( U i , V i , ¯ V i , ˜ V i ) → ( U, V , ¯ V , ˜ V ) as i → ∞ in M D γ [0 , T ] . (5.69) Finally , b y (5.6 9) and th e similar pr o cedure as used for Theorem 5. 2.1 in p ages 6 8-71 of ∅ ksendal [39], w e can complete the p ro of of Lemma 5.3.  5.2 Pro of of Theorem 2.1 By com binin g Lemmas 5.1-5.3, we can reac h a pro of for Theorem 2.1 .  5.3 Pro of of Theorem 2.2 First, w e consider a real-v alued s y s tem corresp ondin g to the case that τ = T , w h ose pro of is along the line of the one for Lemma 5.3. More p recisely , for any giv en num b er sequence γ = { γ D c , c = 0 , 1 , 2 , ... } with γ D c ∈ R , replace the norm for the Banac h s p ace M D γ [0 , T ] deﬁned in (5.46) by the one    ( U, V , ¯ V , ˜ V )    2 M D γ ≡ ∞ X c =0 ξ ( c )    ( U, V , ¯ V , ˜ V )    2 M D c γ D c ,c , (5.70) for an y giv en ( U, V , ¯ V , ˜ V ) in this space, where    ( U, V , ¯ V , ˜ V )    2 M D c γ D c = E " sup 0 ≤ t ≤ T k U ( t ) k 2 C c ( D c ,r ) e 2 γ D c t # 42 + E " sup 0 ≤ t ≤ T k V ( t ) k 2 C c ( D c ,q ) e 2 γ D c t # + E  Z T 0   ¯ V ( t )   2 C c ( D c ,q d ) e 2 γ D c t dt  + E  Z T 0    ˜ V ( t )    2 ν,c e 2 γ D c t dt  . Then, it follo ws from the similar argument u s ed f or (5.66) in the pro of of Lemma 5.3 that ( U 1 ( · , x ) , V 1 ( · , x ) , ¯ V 1 ( · , x ) , ˜ V 1 ( · , x, z )) ∈ ¯ Q 2 F ([0 , T ] × D ) with ( U 0 , V 0 , ¯ V 0 , ˜ V 0 ) = (0 , 0 , 0 , 0), wh ere ( U 1 , V 1 , ¯ V 1 , ˜ V 1 ) is deﬁned through (5.2) in Lemma 5.1. F urtherm ore, o v er eac h D c with c ∈ { 0 , 1 , ... } , we ha v e that    (∆ U i , ∆ V i , ∆ ¯ V i , ∆ ˜ V i )    2 M D γ (5.71) ≤ e k δ     (∆ U i − 1 , ∆ V i − 1 , ∆ ¯ V i − 1 , ∆ ˜ V i − 1 )    2 M D γ +    (∆ U i − 2 , ∆ V i − 2 , ∆ ¯ V i − 2 , ∆ ˜ V i − 2 )    2 M D γ  , where δ is a constan t that can b e determined b y suitably choosing a num b er sequ ence γ s u c h that γ D 0 < γ D 1 < ... and 0 < √ e k δ / (1 − 2 √ e k δ ) < 1 (note that γ D c ma y dep end on b oth D c and c for eac h c ∈ { 0 , 1 , ... } ). Thus, it f ollo ws from (5.71) that the remaining justiﬁcation for Theorem 2.2 can b e conducted along the line of pro of for Th eorem 2.1 . Second, w e consid er a real-v alued system corresp onding to the case that τ is a general stopping time. The p ro of f or this case ca n b e acc omplished b y exte ndin g th e p ro of c orre- sp ond in g to τ = T via the tec hn iques develo p ed in Dai [16, 20] for b oth forw ard and b ac kward SDEs, and the related discussions in Y ong and Zhou [52]. Third, b y direct generalizing the d iscussion concerning the real-v alued system to complex- v alued s ystem, w e reac h a pro of for Theorem 2.2.  6 Pro ofs of Theorem 3.1 and T heorem 3.2 T o pr o vid e the pro ofs for Th eorem 3.1 and T heorem 4.1, we ﬁrst recall the Sko roho d p roblem and study some related prop erties. 6.1 T he Sk oroho d Problem Let D ([0 , T ] , R b ) with b ∈ { p, 2 p } b e the sp ace of all functions z : [0 , T ] → R b that are r igh t- con tinuous with left limits and are endo wed with Skorohod top ology (see, e.g., Billingsley [3], Jaco d and Shir y aev [28]). Then, w e can introd uce th e Skorohod pr oblem as follo ws. 43 Deﬁnition 6.1 (The Skor oho d pr oblem). Given z ∈ D ([0 , T ] , R p ) with z (0) ∈ D , a ( D , R ) - r e gulation of z over [0,T] i s a p air ( x, y ) ∈ D ([0 , T ] , D ) × D ([0 , T ] , R b + ) such tha t x ( t ) = z ( t ) + Ry ( t ) for al l t ∈ [0 , T ] , wher e, for e ach i ∈ { 1 , ..., b } , 1. y i (0) = 0 , 2. y i is nonde cr e asing, 3. y i c an incr e ase only at a time t ∈ [0 , T ] with x ( t ) ∈ F i . F urtherm ore, w e deﬁn e the modu lus of co ntin uit y with resp ect to a function z ( · ) ∈ D ([0 , T ] , R b ) and a r eal num b er δ > 0 by w ( z , δ, T ) ≡ in f t l max l Osc ( z , [ t l − 1 , t l )) , (6.1) where the in ﬁ m um tak es o ver all the ﬁnite sets { t l } of p oin ts satisfying 0 = t 0 < t 1 < ... < t m = T and t l − t l − 1 > δ for l = 1 , ..., m , and Osc( z , [ t l − 1 , t l ]) = sup t 1 ≤ s ≤ t ≤ t 2 k z ( t ) − z ( s ) k . (6.2) Then, w e hav e the follo w ing lemma. Lemma 6.1 Supp ose that the r eﬂe ction matrix R in Deﬁnition satisﬁes the c ompletely- S c ondition. Then, any ( D , R ) -r e gulation ( x, y ) of z ∈ D ([0 , T ] , R p ) with z (0) ∈ D satisﬁes the oscil lation ine quality over [ t 1 , t 2 ] with t 1 , t 2 ∈ [0 , T ] Osc ( x, [ t 1 , t 2 ]) ≤ κ Osc ( z , [ t 1 , t 2 ]) , (6.3) Osc ( y , [ t 1 , t 2 ]) ≤ κ Osc ( z , [ t 1 , t 2 ]) , (6.4) wher e κ is some nonne gative c onstant dep ending only on the inwar d normal ve ctor N and the r eﬂe ction matrix R . Pr oof. F or eac h t ∈ [ t 1 , t 2 ], deﬁ ne ∆ z ( t ) ≡ z ( t ) − z ( t − ) , (6.5) ∆ x ( t ) ≡ x ( t ) − x ( t − ) , (6.6) ∆ y ( t ) ≡ y ( t ) − y ( t − ) . (6.7) Since th e reﬂection matrix R satisﬁes the completely- S condition, it is easy to chec k that the linear complemen tarit y p roblem (LCP) ∆ x ( t ) = ∆ z ( t ) + R ∆ y ( t ) , ∆ x ( t ) ∈ D , ∆ y ( t ) ≥ 0 , ∆ x i ( t )∆ y i ( t ) = 0 for i = 1 , ..., p, ( b i − ∆ x i ( t ))∆ y i ( t ) = 0 for i = p + 1 , ..., b, 44 is completely solv able (see also Theorem 2.1 in Mandelbaum [35] for the r elated discussion). F urtherm ore, we can conclude that ∆ y ( t ) ≤ C ∆ z ( t ) (6.8) for some nonnegativ e constan t C dep end ing only on the in ward normal vecto r N and the reﬂection matrix R . Then, the rest of the pro of is th e direct conclusion of the one for Theorem 3.1 in Dai [14] or the one for Theorem 4.2 in Dai and Dai [12].  Lemma 6.2 Assume that ( x n , y n ) → ( x, y ) along n ∈ { 1 , 2 , ... } in D ([0 , T ] , R p ) × D ([0 , T ] , R b ) and y n ( · ) is of b ounde d variation for e ach n ∈ { 1 , 2 , ... } . F urtherm or e, supp ose that Z t 0 f ( x n ( s )) dy n ( s ) = 0 (6.9) for al l n ∈ { 1 , 2 , ... } and e ach t ∈ [0 , T ] , wh er e f ∈ C b ([0 , T ] , R b ) is a b -dimensional b ounde d ve ctor func tion. Then, for e ach t ∈ [0 , T ] , we have tha t Z t 0 f ( x ( s )) dy ( s ) = 0 . (6.10) Pr oof. It follo ws from the deﬁnition in pages 123- 124 of Billingsley [3] or Theorem 1.14 in page 328 of Jaco d and S hirya ev [28] that th er e is a sequence { γ n , n ∈ { 1 , 2 , ... }} of contin uous and strictly in cr easing functions mapping from [0 , T ] → [0 , T ] with γ n (0) = 0 and γ n ( T ) = T suc h that sup t ∈ [0 ,T ] | γ n ( t ) − t | → 0 , (6.11) sup t ∈ [0 ,T ] | ( x n , y n )( γ n ( t )) − ( x, y )( t ) | → 0 . (6.12) Then, b y the unif orm con v ergence in (6.11)-(6.12) and th e condition in (6.9), w e k n o w that Z t 0 f ( x ( s )) dy ( s ) = lim n →∞ Z t 0 f ( x n ( γ n ( s ))) dy n ( γ n ( s )) = lim n →∞ Z γ − 1 n ( t ) 0 f ( x n ( u )) dy n ( u ) = 0 , where γ − 1 n ( · ) is the inv erse function of γ n ( · ) for eac h n ∈ { 1 , 2 , ... } . Hence, w e complete the pro of of Lemma 6.2.  6.2 Pro of of Theorem 3.1 W e d ivide the pro of of the theorem into f our parts: Part A (Existence, Uniqueness), P art B, P art C, and Pa rt D, whic h corresp ond to d iﬀerent b oundary reﬂection conditions. 45 P art A (Existe nce). W e co nsider the c ase that L ( t, ω ) app eared in (3.1)-(3.2) is a constan t and b oth of the f orward and the bac kward SDEs hav e reﬂection b ound aries. In this case, we n eed to pro v e the claim that there is an adapted w eak solution (( X , Y ) , ( V , ¯ V , ˜ V , F )) to the sy s tem in (1.3). In fac t, for a p ositiv e in teger b , let D 2 F ([0 , T ] , R b ) b e the space of R b -v alued a nd {F t } - adapted p ro cesses with sample paths in D ([0 , T ] , R b ). F urthermore, eac h Y ∈ D 2 F ([0 , T ] , R b ) is sq u are-in tegrable in th e sense that E  Z T 0 k Y ( t ) k 2 dt  < ∞ . (6 .13) In addition, w e use D 2 F ,p ([0 , T ] , R b ) to d en ote the corresp ond ing p r edictable s p ace. Then, for a giv en n ∈ { 1 , 2 , ... } and a 4-tuple ( X n , V n , ¯ V n , ˜ V n ) ∈ D 2 F ([0 , T ] , R p ) × D 2 F ([0 , T ] , R q ) × D 2 F ,p ([0 , T ] , R q × d ) (6.14) × D 2 F ,p ([0 , T ] × R h + , R q × h ) with X n (0) ∈ D and V n ( T ) ∈ ¯ D , w e ha v e the follo wing observ ation. By the study concerning the con tinuous dynamic co mplementa rit y problem (DCP) in Bernard and El K harroubi [2] (see also the relate d discus s ions in Ma ndelbaum [35], Reiman and Williams [47]), Th eorem 2.1 (and its p ro of ) in th e cur ren t p ap er, there is a 6-tuple (( X n +1 , Y n +1 ) , ( V n +1 , ¯ V n +1 , ˜ V n +1 , F n +1 )) ∈ D 2 F ([0 , T ] , R p ) × D 2 F ([0 , T ] , R b ) × D 2 F ([0 , T ] , R q ) × D 2 F ,p ([0 , T ] , R q × d ) × D 2 F ,p ([0 , T ] × Z h , R q × h ) × D 2 F ([0 , T ] , R q × ¯ b ) for eac h n ∈ { 1 , 2 , ... } , satisfying th e prop erties along eac h samp le path: X n +1 ( t ) = X (0) + Z n ( t ) + RY n +1 ( t ) ∈ D , (6.15) with Z n ( t ) = Z n 1 ( t ) + Z n 2 ( t ) , Z n 1 ( t ) = Z t 0 b ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , · ) , u ( s − , X n ( s − ) , · ) ds Z n 2 ( t ) = Z t 0 σ ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , · ) , u ( t, X n ( s − )) , z , · ) dW ( s ) + Z t 0 Z Z h η ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , · ) , u ( s − , X n ( s − )) , z , · ) ˜ N ( ds, dz ); and V n +1 ( t ) = H ( X n ( T )) − S F n ( T ) + U n ( t ) + S F n +1 ( t ) ∈ ¯ D , (6.16) 46 with U n ( t ) = U n 1 ( t ) − U n 2 ( t ) − U n 3 ( t ) , where, U n 1 ( t ) = Z T t c ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , · ) , u ( s − , X n ( s − ) , · ) ds, U n 2 ( t ) = Z T t  α ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , · ) , u ( s − , X n ( s − )) , · ) − ¯ V n ( s − )  dW ( s ) + Z T t Z Z h  ζ ( s − , X n ( s − ) , V n ( s − ) , ¯ V n ( s − ) , ˜ V n ( s − , z ) , u ( s − , X n ( s − )) , z , · ) − ˜ V n ( s − , z )  ˜ N ( ds, dz ) , U n 3 ( t ) = Z T t ¯ V n +1 ( s − ) dW ( s ) + Z T t Z Z h ˜ V n +1 ( s − , z ) ˜ N ( ds, dz ) . F urtherm ore, ( X n +1 , Y n +1 ) satisﬁes the prop ert y (3) in Deﬁnition 4.4. I n other w ords, Y n +1 is a b -dimensional {F t } -adapted pro cess such that th e i th comp onent Y n +1 i of Y n +1 for eac h i ∈ { 1 , ..., b } P -a. s. h as the pr op erties that Y n +1 i (0) = 0, Y n +1 i is non-decreasing, and Y n +1 i can increase only when X n +1 is on the b oun dary face D i , i.e., Z t 0 I D i ( X n +1 ( s )) d Y n +1 i ( s ) = Y n +1 i ( t ) for all t ≥ 0 . (6.17) Similarly , ( V n +1 , F n +1 ) also satisﬁes th e prop ert y (3) in Deﬁnition 4.4. More p recisely , F n +1 is a q -dimens ional {F t } -adapted pro cess such that th e i th comp onent F n +1 i of F n +1 for eac h i ∈ { 1 , ..., ¯ b } P -a.s. h as the pr op erties that F n +1 i (0) = 0, F n +1 i is non-decreasing, and F n +1 i can increase only when V n +1 is on the b oundary face ¯ D i , i.e., Z t 0 I ¯ D i ( V n +1 ( s )) dF n +1 i ( s ) = F n +1 i ( t ) for all t ≥ 0 . (6.18) Next, w e p ro ve that th e follo wing sequen ce of sto c h astic pro cesses along n ∈ { 1 , 2 , ... } , Ξ n = (( X n +1 , Y n +1 ) , ( V n +1 , ¯ V n +1 , ˜ V n +1 , F n +1 )) , ( X 1 , V 1 , ¯ V 1 , ˜ V 1 ) = 0 , (6.19) is r elativ ely co mpact in the Sk oroho d top ology o ve r th e sp ace P [0 , T ] ≡ D 2 F ([0 , T ] , R p ) × D 2 F ([0 , T ] , R b ) (6.20) × D 2 F ([0 , T ] , R q ) × D 2 F ,p ([0 , T ] , R q × d ) × D 2 F ,p ([0 , T ] × Z h , R q × h ) × D 2 F ([0 , T ] , R q × ¯ b ) . Along the line of Dai [14, 18], Dai and Dai [12], and b y Corollary 7.4 in page 129 of Ethier and Kur tz [23], it suﬃces to pro v e the follo wing t wo conditions to b e true: First, for eac h ǫ > 0 and r ational t > 0, there is a constant C ( ǫ, t ) su c h that lim inf n →∞ P n k Ξ n k 2 ≤ C ( ǫ, t ) o ≥ 1 − ǫ ; (6.21) 47 Second, for eac h ǫ > 0 and T > 0, there is a constant δ > 0 su c h that lim sup n →∞ P { w (Ξ n , δ, T ) ≥ ǫ } ≤ ǫ. (6.22) T o p ro v e the tw o conditions stated in (6.2 1) and (6.22), w e ﬁrst deﬁne the norm alo ng eac h sample path k f k [ a,b ] = sup a ≤ t ≤ b k f ( t ) k for eac h f ∈ { X n , Z n , U n , ( V n , ¯ V n , ˜ V n ) } an d eac h a, b ∈ [0 , T ]. T hen, w e introdu ce the space for some constant γ > 0 that will b e chosen and explained in the follo wing pr o of, Q γ [0 , T ] ≡ D 2 F ([0 , T ] , R p ) × D 2 F ([0 , T ] , R q ) × D 2 F ,p ([0 , T ] , R q × d ) (6.23) × D 2 F ,p ([0 , T ] × Z h , R q × h ) endo w ed with the n orm    ( X, V , ¯ V , ˜ V )    2 Q γ [0 ,T ] (6.24) ≡ E " sup t ∈ [0 ,T ]  k X ( t ) k 2 + k V ( t ) k 2  e 2 γ t # + E  Z T 0   ¯ V ( t )   2 e 2 γ t dt  + E  Z T 0    ˜ V ( t, · )    2 ν e 2 γ t dt  for ea c h ( X, V , ¯ V , ˜ V ) ∈ Q γ [0 , T ]. Thus, by Lemma 6.1, there is a p ositiv e constan t C 1 suc h that   ( X n +1 , Y n +1 )( t )   (6.25) ≤   ( X n +1 , Y n +1 )(0)   + κ Osc( Z n , [0 , T ]) ≤ C 1  k X (0) k + k Z n k [0 ,T ]  , and    ( V n +1 , ¯ V n +1 , ˜ V n +1 ( · ) , F n +1 )( t )    (6.26) ≤    ( V n +1 , ¯ V n +1 , ˜ V n +1 ( · ))( t )    +   F n +1 ( t )   ≤    ( V n +1 , ¯ V n +1 , ˜ V n +1 ( · ))( T )    +   F n +1 (0)   + 2 κ Osc( U n , [0 , T ]) ≤ ¯ C 1  k ( V n ( T ) k + k F n ( T ) k + k U n k [0 ,T ]  ≤ ¯ C 2  1 + k X n ( T ) k +   U n − 1   [0 ,T ] + k U n k [0 ,T ]  , ≤ C 1  1 + k X (0) k +   Z n − 1   [0 ,T ] +   U n − 1   [0 ,T ] + k U n k [0 ,T ]  , 48 where, ¯ C 1 and ¯ C 2 are some nonnegativ e constan ts. F urtherm ore, we ha ve tak en ( ¯ V n +1 , ˜ V n +1 ( · ))( T ) = 0 in th e thir d equalit y of (6.28) since the uniqu eness for the Martingale represent ation is in the sense of up to sets of measur e zero in ( t, ω ) (see, e.g., Theorem 4.3.4 in page 53 of ∅ ksendal [39] and Theorem 5.3.5 in page 266 of Applebaum [1]). Th us, for eac h n ∈ { 1 , 2 , ... } , the give n linear growth constan t L ≥ 0 in ( 3.1), and any constan t K > LT , it f ollo ws from the Mark o v’s inequalit y that P {k Z n 1 k T ≥ K } ≤ LT K − LT E     ( X n , V n , ¯ V n , ˜ V n ( · ))    [0 ,T ]  . (6.27) F urtherm ore, b y Lemma 4.2.8 in page 201 of Applebaum [1] (or related theorem in page 20 of Gihman and Sko roho d [24]) and the linear gro wth condition, w e kno w that P {k Z n 2 k T ≥ K } ≤ ¯ K K 2 + L 2 T ¯ K − L 2 T E     ( X n , V n , ¯ V n , ˜ V n ( · ))    2 [0 ,T ]  (6.28) for all n onnegativ e constan t ¯ K > L 2 T . In addition, similar to th e illustration of (6.27), we ha v e that P {k U n 1 k T ≥ K } ≤ 1 K − LT E     ( X n , V n , ¯ V n , ˜ V n ( · ))    [0 ,T ]  . ( 6.29) Next, by the s imilar demonstration for (6.28) and the linear gro w th condition, we know that P {k U n 2 k T ≥ K } ≤ ¯ K K 2 + L 2 T ¯ K − L 2 T E     ( X n , V n , ¯ V n , ˜ V n ( · ))    2 [0 ,T ]  . (6.30) F urtherm ore, by the pro of of Theorem 2.1, w e ha ve that P {k U n 3 k T ≥ K } ≤ ¯ K K 2 + ¯ K 1 T ( ¯ K − L 2 T ) 2 (6.31) + ¯ K 2 ( ¯ K − L 2 T ) 2 E     ( X n , V n , ¯ V n , ˜ V n )    2 Q γ [0 ,T ]  for some n on n egativ e constan ts ¯ K 1 and ¯ K 2 . T herefore, for ea c h giv en ǫ > 0, it follo ws f rom (6.27)-(6.31), suitably c hosen constan ts K and ¯ K , and the initial condition in (6.19) that there is a nonn egativ e constan t C suc h that inf n P {k Ξ n ( t ) k ≤ C , 0 ≤ t ≤ T } (6.32) ≥ inf n min  P    ( X n +1 , Y n +1 )( t )   ≤ C, 0 ≤ t ≤ T  , P n    ( V n +1 , ¯ V n +1 , ˜ V n +1 , F n +1 )( t )    ≤ C, 0 ≤ t ≤ T oo ≥ 1 − ǫ. Th us, the cond ition in (6.21) is satisﬁed b y the sequence of { Ξ n } . 49 No w, for any t ∈ [0 , T ], it f ollo ws from the pro of of Prop osition 18 for a BSDE with jumps in Dai [16] and Lemma 6.1 that    ( U n , ¯ V n , ˜ V n )    2 Q γ [ t,T ] (6.33) ≤ K γ  2 L 2 ( T − t ) +    ( X n − 1 , V n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [ t,T ]  ≤ K γ  2 L 2 ( T − t ) + e 2 γ T E h   V n − 1   2 [ t,T ] i + e 2 γ T Z T t E h   X n − 1   2 [0 ,s ] i ds  + K γ    ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [ t,T ] , where, K γ < 1 dep ending only on L , T , d , and h for some suitable c hosen γ > 0. Th us, b y Lemma 6.1 , th e It ˆ o ’s isometry f ormula, and (6.33), we ha v e that E h k V n k 2 [ t,T ] i (6.34) ≤ ¯ K 1  E  k V n ( T ) k 2  + E  k F n − 1 ( T ) k 2  + κ 2 E  Osc( U n − 1 , [ t, T ]) 2  ≤ K 1  1 + E h k X n k 2 [0 ,T ] i + κ 2 E  Osc( U n − 2 , [0 , T ]) 2  + κ 2 E  Osc( U n − 1 , [ t, T ]) 2   ≤ K 1  1 + 24 κ 2 L 2 T 2 + 24 κ 2 L 2 ( T − t ) 2  + K 1 E h k X n k 2 [0 ,T ] i +24 K 1 κ 2 L 2 T  Z T 0 E h   X n − 2   2 [0 ,s ] i ds + E h   V n − 2   2 [0 ,T ] i  +24 K 1 κ 2 L 2 ( T − t )  Z T t E h   X n − 1   2 [0 ,s ] i ds + E h   V n − 1   2 [ t,T ] i  +24 K 1 κ 2 L 2 T    ( U n − 2 , ¯ V n − 2 , ˜ V n − 2 )    2 Q γ [0 ,T ] +4 K 1 κ 2    ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [0 ,T ] +24 K 1 κ 2 L 2 ( T − t )    ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [ t,T ] +4 K 1 κ 2    ( U n , ¯ V n , ˜ V n )    2 Q γ [ t,T ] ≤ K 3 + K 2  Z T 0 E h   X n − 1   2 [0 ,s ] i ds +    ( U n − 2 , ¯ V n − 2 , ˜ V n − 2 )    2 Q γ [0 ,T ] +    ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [0 ,T ] +    ( U n , ¯ V n , ˜ V n )    2 Q γ [0 ,T ]  , where, K i for i ∈ { 1 , 2 , 3 } are some nonnegativ e constants dep ending only on T , L , κ , and E  k V ( T ) k 2  . F urthermore, for any t ∈ [0 , T ], w e ha v e that E h k X n k 2 [0 ,t ] i ≤ 2 E h k X (0) k 2 i + 2 κ 2 E  Osc( Z n − 1 , [0 , t ]) 2  (6.35) ≤ 2 E h k X (0) k 2 i + 6 κ 2 L 2 t 2 50 +6 κ 2 L 2 t  Z t 0 E h   X n − 1   2 [0 ,s ] i ds + E h   V n − 1   2 [0 ,T ] i  +6 κ 2 L 2 t    ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [0 ,T ] ≤ 2 E h k X (0) k 2 i + 12 κ 4 L 2 t 2 + 6 κ 2 L 2 tE    V 2 ( T )    +6 κ 2 L 2 t Z t 0 E h   X n − 1   2 [0 ,s ] i ds +6 κ 2 L 2 t  1 + 2 κ 2     ( U n − 1 , ¯ V n − 1 , ˜ V n − 1 )    2 Q γ [0 ,T ] . Therefore, for an y ǫ > 0 and a constan t δ > 0, consider a ﬁnite set { t l } of p oin ts satisfying 0 = t 0 < t 1 < ... < t m = T and t l − t l − 1 = δ < ǫ /L with l ∈ { 1 , ...m } . It follo ws from (6.19), (6.33)-(6.35), and the similar explanation for (6.27) that P { w ( Z n 1 , δ, T ) ≥ ǫ } (6.36) ≤ 3 L 2 δ ( ǫ − Lδ ) 2  E h k X n k 2 [0 ,T ] + k V n k 2 [0 ,T ] i +    ( U n , ¯ V n , ˜ V n )    2 Q γ [0 ,T ]  ≤ 3 L 2 δ ( ǫ − Lδ ) 2 A 0 + n X k =1 A k +1 1 T k +1 ( k + 1)!  1 + K k γ  + A 2 n X k =1 K k γ ! , where A 0 , A 1 , and A 2 are some constan ts dep ending only on L , T , d , and h . F urther m ore, b y Lemma 4.2. 8 in page 201 of Applebaum [1] (or related theorem in page 20 of Gihm an an d Sk oroho d [24]) and the linear gro wth condition, w e kno w that P { w ( Z n 2 , δ, T ) ≥ ǫ } (6.37) ≤ ¯ ǫ ǫ 2 + 3 L 2 ¯ ǫ − 3 L 2 δ  δ E h k X n k 2 T i + δ E h k V n k 2 T i + E     ( U n , ¯ V n , ˜ V n )    2 Q γ [0 ,T ]  ≤ ¯ ǫ ǫ 2 + 3 L 2 ¯ ǫ − 3 L 2 δ δ A 0 + n X k =1 A k +1 1 T k +1 ( k + 1)!  1 + K k γ  + A 2 n X k =1 K k γ ! + A 3 n X k =1 K k γ ! for all n onnegativ e constant ¯ ǫ > 3 L 2 δ , wh ere A 3 is some constant dep ending only on L , T , d , and h . Similarly , there are some constan ts B 0 , B 1 , B 2 , and B 3 dep end ing only on L , T , d , and h suc h that P { w ( U n 1 , δ, T ) ≥ ǫ } (6.38) ≤ 3 L 2 δ ( ǫ − Lδ ) 2 B 0 + n X k =1 B k +1 1 T k +1 ( k + 1)!  1 + K k γ  + B 2 n X k =1 K k γ ! , and P { w ( Z n 2 , δ, T ) ≥ ǫ } (6.39) ≤ ¯ ǫ ǫ 2 + 3 L 2 ¯ ǫ − 3 L 2 δ δ B 0 + n X k =1 B k +1 1 T k +1 ( k + 1)!  1 + K k γ  + B 2 n X k =1 K k γ ! + B 3 n X k =1 K k γ ! . 51 Hence, for eac h giv en ǫ > 0, it follo ws from (6.36)-(6.39) and suitably c hosen constants ¯ ǫ , δ , and γ that lim sup n →∞ P { w (Ξ n ) , δ, T ) ≥ ǫ } ≤ ǫ. (6.40 ) Th us, the condition in (6.22) is true for the sequence of { Ξ n } . Hence, b y (6.28), (6.40), and Corollary 7.4 in p age 129 of Ethier and Kurtz [23], this sequence is relativ ely compact. Th ere- fore, t here is a su bsequence of { Ξ n } that con v erges w eakly to Ξ ≡ (( X , Z , Y ) , ( V , ¯ V , ˜ V , F )) o ver the sp ace P [0 , T ]. F or con v enience, w e supp ose that the sub sequence is the sequence itself, i.e., Ξ n ⇒ Ξ . (6.41) Then, by the Skorohod represen tation theorem (see, e.g., Theorem 1.8 in page 102 of Ethier and Kurtz [23]), we can assume that the con v ergence in (6.41) is a.s. in the Skorohod top ology . Th us, by the clai m (a) in Th eorem 1.14 (or the claim ( a) in Prop osition 2.1) of Jaco d and Shiryae v [28] a nd th e facts that Y n +1 (0) = 0 and Y n +1 is nond ecreasing, we can conclude that Y (0) = 0 and Y is nondecreasing. F urthermore, b y Lemma 6.2 and (6.17) Z t 0 I D i ( X ( s )) d Y i ( s ) = Y i ( t ) for all t ≥ 0 , i ∈ { 1 , ..., b } . (6.42) Similarly , we kno w that F (0) = 0, F is non-decreasing, and Z t 0 I ¯ D i ( V ( s )) dF i ( s ) = F i ( t ) for all t ≥ 0 , i ∈ { 1 , ..., ¯ b } . (6.43) Therefore, b y the Lipsc hitz condition in (3 .2), w e know that (( X , Y ) , ( V , ¯ V , ˜ V , F )) satisﬁes the FB-SDEs in (1.3) a.s. Th us, b y the Sk oroho d r epresen tation theorem aga in, it is a w eak solution to the FB-SDEs in (1.3). P art A (Uniqueness). Assum e th at (( X, Y ) , ( V , ¯ V , ˜ V , F )) is a w eak so lution t o the FB-SDEs in (1.3 ). T o pro ve its un iqueness, w e i ntrodu ce some additional notatio ns. Let D ∅ = D , ¯ D ∅ = ¯ D , and d eﬁne D K ≡ ∩ i ∈ K D i , ¯ D ¯ K ≡ ∩ i ∈ ¯ K ¯ D i (6.44) for eac h ∅ 6 = K ⊂ { 1 , ..., b } and eac h ∅ 6 = ¯ K ⊂ { 1 , ..., ¯ b } . In the s equel, w e call a set K ∈ { 1 , ..., b } “ maximal” if K 6 = ∅ , D K 6 = ∅ , a nd D K 6 = D ˜ K for an y ˜ K ⊃ K such that ˜ K 6 = K . Similarly , w e c an deﬁne the maximal set corresp onding to a set ¯ K ∈ { 1 , ..., ¯ b } . F urtherm ore, let d ( x, D K ) and d ( ¯ x, ¯ D ¯ K ) resp ectiv ely denote the Euclidean d istance b et ween x and D K for a p oin t x ∈ D and the Euclidean d istance b etw een a p oin t ¯ x ∈ ¯ D and ¯ D ¯ K . Then, it follo ws from Lemma 3.2 in Dai [14] or Lemma B.1 in Dai and Willi ams [13] that there exist t wo constan ts C ≥ 1 and ¯ C ≥ 1 suc h that d ( x, D K ) ≤ C X i ∈ K ( n i · x − b i ) , ¯ d ( ¯ x, ¯ D ¯ K ) ≤ ¯ C X i ∈ ¯ K ( ¯ n i · ¯ x − ¯ b i ) . (6.45 ) 52 No w, for eac h ǫ ≥ 0, K ∈ { 1 , ..., b } , and ¯ K ∈ { 1 , ..., ¯ b } (including the empty set), w e let D ǫ K ≡ { x ∈ R q : 0 ≤ n i · x − b i ≤ C ǫ for all i ∈ K, (6.46) n i · x − b i > ǫ for all i ∈ { 1 , ..., b } \ K } , ¯ D ǫ ¯ K ≡  ¯ x ∈ R q : 0 ≤ ¯ n i · ¯ x − ¯ b i ≤ ¯ C ǫ for all i ∈ ¯ K , (6.47) ¯ n i · ¯ x − ¯ b i > ǫ for all i ∈ { 1 , ..., ¯ b } \ ¯ K  , where C ǫ = C pǫ and ¯ C ǫ = ¯ C q ǫ . Th u s, by Lemmas 4.1-4.2 in Dai and Williams [13], we kno w that D = ∪ K ∈G D ǫ K , ¯ D = ∪ ¯ K ∈ ¯ G ¯ D ǫ ¯ K , (6.48) where, G is the collection of subs ets of { 1 , ..., b } consisting of all maximal sets in { 1 , ..., b } and ¯ G is deﬁned in the same w a y in terms of su bsets of { 1 , ..., ¯ b } . F or conv enience, we order the sets in G and ¯ G . Then, w e can deﬁne a sequence of 3-dimensional p oint s { ( r n , ¯ r n , τ n ) , n ∈ { 1 , 2 , ... }} with τ 0 = 0 by induction. In fact, since (( X, Y ) , ( V , ¯ V , ˜ V , F )) is a weak s olution to the FB-SDEs in (1.3), b oth X (0) and V (0) are deﬁn ed. Thus, if ( r 1 , ¯ r 1 ) is the ﬁrst K × ¯ K ∈ { 1 , ..., b } × { 1 , ..., ¯ b } su c h that ( x, ¯ x ) ∈ D ǫ r 1 × ¯ D ǫ ¯ r 1 , w e let τ 1 = inf  t ≥ 0 : ( X ( t ) , V ( t )) / ∈ D ǫ r 1 × ¯ D ǫ ¯ r 1  . (6 .49) F urtherm ore, if ( r n , ¯ r n , τ n ) has b een deﬁned on { τ n < ∞} , w e let ( r n +1 , ¯ r n +1 ) b e the ﬁrst K × ¯ K ∈ G × ¯ G su ch that ( X ( τ n ) , V ( τ n )) ∈ D ǫ K × ¯ D ǫ ¯ K . Then, w e can deﬁne τ n +1 = inf n t ≥ τ n : ( X ( t ) , V ( t )) / ∈ D ǫ r n +1 × ¯ D ǫ ¯ r n +1 o . (6.50) On { τ n = + ∞} , w e deﬁ n e r n +1 = r n , ¯ r n +1 = ¯ r n , and τ n +1 = τ n . Due to the r igh t-conti nuit y of the sample paths of solution ( X, V ) by the r elated prop ert y of L´ evy pro cess dr iv en sto c h astic in tegral (see , e.g., Theorem 4.2 .12 in page 204 of Applebaum [1]), { τ n } is a nond ecreasing sequence of {F t } -stopping times, satisfying τ n → ∞ a.s. as n → ∞ . Hence, it suﬃ ces to pro ve the w eak uniqueness of (( X, Y ) , ( V , ¯ V , ˜ V , F ))( · ∧ τ n ) for eac h n . Note that b oth D ǫ r n and ¯ D ǫ ¯ r n for eac h n are subsets of cones. Th us, without loss of generalit y , w e assume that b oth D and ¯ D are cones. Therefore, w e can pro ve the w eak uniqueness by induction in terms of the n u m b ers of b oundary faces of D and ¯ D . In fact, for the case that b = ¯ b = 1, it follo ws fr om the uniqueness of the Skorohod mapping giv en b y Lemma 3.1 in Dai [14] or Lemma 4.5 in Dai and Dai [12] that the we ak uniqueness is true. No w, w e sup p ose that the we ak uniqu eness is true for the c ase that b + ¯ b = m ≥ 2 with b ≥ 1 and ¯ b ≥ 1. Then, w e can prov e the case for b + ¯ b = m + 1. In this case, we need to consid er tw o folds indexed b y t wo pairs of ( b + 1 , ¯ b ) and ( b, ¯ b + 1). Both of the folds can b e pro v ed by the similar discus sion for Theorem 5.4 in Dai and Williams [13 ]. Therefore, w e ﬁn ish the pro of of w eak uniqueness. 53 P art B. W e consider the case that L ( t, ω ) app eared in (3.1)-(3.2) is a constant and the sp ectral radii of S and eac h p × p sub-principal matrix of N ′ R are strictly less than one. In this case, w e need to pr o ve that there is a u nique strong adapted solution (( X , Y ) , ( V , ¯ V , ˜ V , F )) to the sy s tem of in (1.3). In fact, it follo ws from the discussions in Reiman and Harrison [27], Dai [18], Lemma 7.1 and Theorem 7.2 in p ages 164-1 65 of Chen and Y ao [11] that there exist t wo Lipsc hitz con tinuous mappings Φ and Ψ suc h that ( X n +1 , Y n +1 ) = Φ( Z n ) (6.51) ( V n +1 , F n +1 ) = Ψ( U n ) (6.52) for eac h n ∈ { 1 , 2 , ... } . Then, it follo ws fr om (6.51)-(6.52 ), the related estimates in P art A, and the con ven tional Picard’s iterativ e metho d, we can reac h a pr o of for the claim in Part B. P art C. W e consider the case that L ( t, ω ) app eared in (3.1)-(3.2) is a constan t and b oth of the SDEs ha ve no reﬂection b oun daries. In this case, we n eed to p r o ve that there is a unique strong adapted s olution (( X , Y ) , ( V , ¯ V , ˜ V , F )) to the system of in (1.3). In fact, by the related estimate s in Pa rt A, th is case can b e pro v ed b y d irectly generaliz ing the con v en tional Picard’s iterativ e metho d. Actually , this case is a sp ecial one of Th eorem 2.1 or Theorem 2.2. P art D. W e consider the case that L ( t, ω ) app eared in (3.1)-(3.2) is a general adapted and mean-squarely in tegrable sto c h astic pro cess. Th e pro ofs co rresp onding to the cases sta ted in Pa rt A, P art B, and Pa rt C can b e accomplished a long the lines of pr o ofs for Lemma 4.1 in Dai [16] associated with a forward SDE under random en vironment and Prop osition 18 in Dai [20] for a bac kward SDE under random envi ronment. The key in the pro ofs is to in tro duce the follo wing sequ ence of {F t } -stopping times, i.e., τ n ≡ inf { t > 0 , k L ( t ) k > n } for eac h n ∈ { 1 , 2 , ... } . ( 6.53) By the condition in (3.6), τ n is n ondecreasing and a.s. tends to inﬁ n it y as n → ∞ . Finally , b y summarizing the cases presented in P art A to P art D, we ﬁnish th e pr o of of Theorem 3.1.  6.3 Pro of of Theorem 3.2 F or a control pro cess u ∗ ∈ C , it follo ws from Theorem 2.1 that the ( r, q + 1)-dimensional FB-SPDEs in (1.1) w ith the partial diﬀeren tial op erators { ¯ L , ¯ J , ¯ I } giv en b y (3.7)-(3.12) and terminal condition in (3.13) indeed admits a w ell-p osed 4-tuple solution ( U ( t, x ), V ( t, x ), ¯ V ( t, x ), ˜ V ( t, x, · )). Th us, substituting ( V ( t ) , ¯ V ( t ) , ˜ V ( t, · )) ≡ ( V ( t, X ( t )) , ¯ V ( t, X ( t )) , ˜ V ( t, X ( t ) , · )) in to the system of coupled FB-SDEs in (1.3), it follo ws fr om Theorem 3.1 that the claims in Theorem 3.2 are true.  54 7 Pro of of Theorem 4.1 The pr o of of p art 1 is the dir ect extension of the single-dimensional case (i.e., p = q = 1) for the related optimal con trol pr ob lem in ∅ ksendal et al. [41]. The p ro of of part 2 can b e d one as follo ws. F or eac h u ∈ C and γ ( t, x ) = β ( t, x ) ≡ 0, it follo w s fr om Theorem 3.2 that the r egulator processes F ( t ) and Y ( t ) exist. S in ce they are nondecreasing with resp ect to time v ariable t , the deriv ativ es dF dt ( t, x ) and d Y dt ( t, x ) exist a.e. in terms of t ime v ariable t alo ng eac h sample path a. s. F urthermore, if eac h q × q sub-prin cipal matrix of ¯ N ′ S and eac h p × p sub-prin cipal matrix of N ′ R are inv ertible, these deriv at iv es are u niquely d etermined owing to the Skorohod mapping. Nev ertheless, if only the general completely- S condition is imp osed, these deriv ativ es are wea kly unique in a probab ility distribu tion sense. In addition, it follo ws from Prop osition 7.1 in Ethier and Kurtz [23] that these deriv ativ es can b e app ro ximated by p olynomials in terms of v ariable x for ea c h giv en t , wh ic h are denoted b y γ ( t, x ) and β ( t, x ). 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Unified Systems of FB-SPDEs/FB-SDEs with Jumps/Skew Reflections and Stochastic Differential Games

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