Stochatic Perrons method and verification without smoothness using viscosity comparison: the linear case

STOCHASTIC PERRON’S METHOD AND VE R IFICA TION WITHOUT SMOOTHNESS USING VISCOSITY COMP ARISON: THE LINE AR CASE ERHAN BA YRAKT AR AND MIHAI S ˆ IRBU Abstract. W e int ro duce a sto c hastic version of the classical Perron’s method to construct vis cosit y solutions to linear paraboli c equations associated to sto- c hastic different ial equations. Using this method, we construct easily tw o vis- cosit y (sub and sup er) solutions that squeeze in b et we en the exp ected pay off. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the sto ch astic differen tial equa- tion. The unique viscosity solution is actually equal to the exp ected pay off. This amounts to a v erification result (Itˆ o’s Lemma) f or non-smo oth viscosit y solutions of the linear parab olic equation. 1. Introduction The b est wa y to approach a F eynman-Ka c eq uation (or a Hamilton- J acobi- Bellman equa tio n in the case of sto chastic co n trol) is to prov e exis tence of a smo oth solution and then use Itˆ o’s Lemma to r elate it to the corres p onding probabilistic representation. In case such a smo o th solution do es not exist, a larg e num ber of results in the litera ture co nsist in taking the exp ected pay off (v alue function) as so ci- ated to a Marko v diffusion a nd then checking the viscosity solution prop erty . Such an a ppr oach ess ent ially us e s the Markov pr op e rty o f the diffusion. If uniquenes s in law of the sto chastic differential equation do es not hold, then a Markov selection is needed to obtain a viscosity so lution this w ay . If, in a ddition, a viscosity compar - ison result holds true (which is a purely ana lytical res ult), then the co nclusion is that the expec ted pay off (v alue function) is actually the unique visco sity solution. On the o ther hand, Ishii [2] refined the classical Perron’s metho d to the case of viscos it y solutions. This amo un ts to a very p ow erful analytic al metho d to con- struct (therefor e proving e x istence) v iscosity solutio ns in very gener al frameworks. How e v er, if one wan ts to co mpare such a viscos it y solution obtained by Perron’s metho d with the exp ected pay-off (v a lue function), then o ne still ne e ds the viscos- it y prop erty for the ex pected pay-off (v alue function). In other words, the pr ogram describ ed in the b eginning still nee ds to be carr ied o ut. Date : October 23, 2018. 2000 Mathematics Subje ct Classific ation. Primary 60G46, 60H30; Secondary 35J88, 35J40. Key wor ds and phr ases. Perron’s method, viscosity solutions, non-smo oth v erification, com- parison pr inciple. The r esearc h of E. Bayrakt ar wa s supported in part b y the National Science F oundation under gran ts DMS 0906257 and D M S 0955463. The research of M. S ˆ ırbu was supported in part by the National Science F oundation under Gran t DMS 0908441. The authors would like to thank Gerard Bruni ck and Gordan ˇ Zitk o viˇ c f or their commen ts. Special thanks go to Ioannis Karatzas for his suggestions that led to an i m pro v ed final v ersion. 1 2 ERHAN BA YRAKT AR AND MI HAI S ˆ IRBU In this no te, we prop ose a st o chastic alternative to Perron’s metho d to con- struct viscosity s olutions, namely Theorem 2 .1. More precisely , we consider the infim um of sto chastic sup er-so lutions or the supremum of sto chastic sub-solutions to a linea r parab olic PDE. By sto chastic sub and s uper -solutions we mean obvious generaliza tions of the seminal notion of sto chastic solution intro duced by Stro o ck and V aradha n [5]. T o the b est o f our knowledge, such a tec hnique do es not ex ist in the liter ature. While this constructio n do es not provide a sto chastic solution, it do es provide a (weaker) viscosity solution. The main adv an tage of o ur metho d is that co mparison b etw e e n such constructed viscosity solutions and the exp ected pay-off (v alue function) becomes tr ivial (see Lemma 2.1), because it is imb e dde d in the sto cha stic definition . In other w ords, one does not need to prov e a n y proper t y for the expected pay-off (v alue function) in order to compar e it to the visco sity solution(s) c o nstructed by sto chastic Perron’s metho d. Using this result, if viscos- it y compar ison holds, then o ne gets that the unique visc osity s olution is actually equal to the ex pected pay-off (v alue function) for fr e e . The unique v is cosity solu- tion is a martingale alo ng any solutio n of the sto chastic differential eq uation, i.e. is a sto chastic solution in the se ns e of Stro o ck and V aradhan [5]. This actually amounts to a verification r esult for non-smo oth viscosity so lutions, where we can use uniqueness of v iscosity solutio ns as a substitute for v erification. In the present note w e illustr ate these ideas in the simplest framew ork of line ar parab olic equations with terminal conditions on the whole sta te space (a particular version o f F eynman- K ac). Howev er, we claim that thess ide a s car r y o ver to muc h more g e neral fra meworks. In particular, other linear ca ses including infinite hor i- zons, running-co sts, exit times or even reflections on the b oundary can b e easily treated in a n identical wa y . Mor e interestingly , we intend to car ry ov er these ide a s to the more imp ortant ca se of Hamilton-Jaco bi-Bellman equa tions as so ciated to sto chastic co n trol and sto chastic games (Isaac ’s e q uations). These more technical details a r e left to future w ork and will b e presented in fo rthcoming pap ers. 2. The s et-up and main resul ts Fix a time interv al T > 0 and for ea ch 0 ≤ s < T and x ∈ R d consider the sto chastic differential equa tion (1)  dX t = b ( t, X t ) dt + σ ( t, X t ) dW t , s ≤ t ≤ T X s = x. W e assume that the co efficients b : [0 , T ] × R d → R d and σ : [0 , T ] × R d → M d,d ′ ( R ) are contin uous. W e also as sume tha t, for each ( s, x ) eq uation (1) has a t least a weak non-ex plo ding solution  ( X s,x t ) s ≤ t ≤ T , ( W s,x t ) s ≤ t ≤ T , Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T  , where the W s,x is a d ′ -dimensional Brownian motion on the sto chastic basis (Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T ) and the filtration ( F s,x t ) s ≤ t ≤ T satisfies the usual co nditions. W e denote by X s,x the non-empty set o f such w eak solutions. It is w ell known, for example fro m [4], that a sufficient condition for the existence of non-explo ding solutions, in addition to contin uit y of the co efficients, is the condition of linear growth: | b ( t, x ) | + | σ ( t, x ) | ≤ C (1 + | x | ) , ( t, x ) ∈ [0 , T ] × R d . STOCHASTIC PERRON’S METHOD 3 W e emphasize tha t we do not as s ume uniqueness in law of the weak solution. Remark 2.1. A ctual ly, in or der to insur e that X s,x is a set in the s ense of ax- iomatic set the ory, one should r estrict to we ak solutions wher e t he pr ob ability sp ac e Ω is an element of a fixe d u n iversal set S of p ossible pr ob abil ity sp ac es. Now, for some fixed b ounde d a nd meas urable function g : R d → R , w e deno te by v ∗ ( s, x ) := inf X s,x E s,x [ g ( X s,x T )] , and v ∗ ( s, x ) := sup X s,x E s,x [ g ( X s,x T )] . W e will call the functions v ∗ and v ∗ the lower a nd the upp er exp ected pay-offs (v alue functions). It is obvious that v ∗ ≤ v ∗ and the tw o functions co inc ide if the sto chastic differential equation (1) has a unique in law weak solution. Remark 2.2. At this stage, we c annot even c onclude that v ∗ and v ∗ ar e me asur able. W e exp ect that the expe c ted pa y-offs (v alue functions) v ∗ and v ∗ be a sso ciated to the following linear PDE: (2)  − u t − L t u = 0 u ( T , x ) = g ( x ) , where the time dependent op erator L t is defined by ( L t u )( x ) = h b ( t, x ) , ∇ u ( t, x ) i + 1 2 T r ( σ ( t, x ) σ T ( t, x ) u xx ( t, x )) , 0 ≤ t < T , x ∈ R d . 2.1. Sto c hasti c P erron’s metho d. Let g : R d → R b e meas urable and b ounded. As mentioned in the Introduction, we now intro duce the sets of sto chastic sup er and sub-s o lutions of the par a bo lic PDE (2 ) in the spirit of [5]. Definition 2.1. The set of sto chastic sup er-solutions of t he p ar ab olic PDE (2 ) , denote d by U + , is the set of functions u : [0 , T ] × R d → R which have the fol lowing pr op erties (i) ar e upp er semic ontinu ous (USC) and b ounde d on [0 , T ] × R d . In addition, they satisfy the terminal c ondition u ( T , x ) ≥ g ( x ) for al l x ∈ R d . (ii) for e ach ( s, x ) ∈ [0 , T ] × R d , and e ach we ak solution  ( X s,x t ) s ≤ t ≤ T , ( W s,x t ) s ≤ t ≤ T , Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T  ∈ X s,x , the pr o c ess ( u ( t, X s,x t )) s ≤ t ≤ T is a sup erm art ingale on (Ω s,x , P s,x ) with r e- sp e ct to the filtr ation ( F s,x t ) s ≤ t ≤ T . Definition 2. 2. The set of sto cha stic sub-solutions of t he p ar ab olic PDE (2) , de- note d by U − , is the set of functions u : [0 , T ] × R d → R which have the fol lowing pr op erties (i) ar e lower semic ontinuous (LSC) and b ounde d on [0 , T ] × R d . In addition, they satisfy the terminal c ondition u ( T , x ) ≤ g ( x ) for al l x ∈ R d . (ii) for e ach ( s, x ) ∈ [0 , T ] × R d , and e ach we ak solution  ( X s,x t ) s ≤ t ≤ T , ( W s,x t ) s ≤ t ≤ T , Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T  ∈ X s,x , the pr o c ess ( u ( t, X s,x t )) s ≤ t ≤ T is a submartingale on (Ω s,x , P s,x ) with r esp e ct to the filtr ation ( F s,x t ) s ≤ t ≤ T . 4 ERHAN BA YRAKT AR AND MI HAI S ˆ IRBU Remark 2 .3. In the Definitions 2.1, 2.2 of U + , U − we do not assume that the pr o c esses ( u ( t, X s,x t )) s ≤ t ≤ T have (RC) right-c ont inous p aths. F or this r e ason, c ar e must b e taken when one tries to apply the Optional Sampling The or em in the form of “a stopp e d martingale is a martingale”. Mor e pr e cisely, such a the or em holds only with r esp e ct to discr ete-value d stopping times. Remark 2.4. Sinc e g is assume d b ounde d, the sets U − and U + ar e e asily se en to b e non-empty. Mor e pr e cisely any c onst ant funct ion u ( t, x ) ≡ k which is an upp er b ound to g ( g ≤ k ) is in U + and any c onstant function u ( t, x ) ≡ k which is a lower b ound to g ( k ≤ g ) is in U − . If one wants t o ac c ount for a lar ger class of functions g than b ou n de d, then the definitions of U − and U + should b e change d appr opriately, and an assumption on n on-emptiness of U − and U + should b e made. Using the prop erties o f sub(sup er)-mar tingales as well as the de finitio n of v ∗ and v ∗ , we eas ily obtain the following result. Lemma 2 .1. F or e ach u ∈ U − and e ach w ∈ U + we have u ≤ v ∗ ≤ v ∗ ≤ w . Using the Remark 2.4 and Lemma 2.1, w e can define v − := sup u ∈U − u ≤ v ∗ ≤ v ∗ ≤ v + := inf w ∈ U + w. Lemma 2 .2. W e have v − ∈ U − , v + ∈ U + . Pr o of. It is well known that an infimum of upper semico n tinous functions is upp er semicontin uous. While we cannot conclude dir ectly that the p oint-wise infimum of sup e rmartingales is a sup ermartinga le (b ecause the set o f super martingales may b e unc ountable , a nd the use of essen t ial infimum would be needed), we c an app eal to Prop osition 4.1 in the App endix and conclude that v + is ac tua lly the po in t-wise infim um of a c ountable set of functions w n ∈ U + , n = 1 , 2 . . . . . Now, the p oint-wise infim um of a c ountable se t of sup ermartinga les is, indeed, a supe rmartingale. The terminal condition fo r v + is satisfied a nd the boundedness follows eas ily s ince g is bo unded so v ∗ is b ounded, and us ing Remar k 2.4 and Lemma 2.1 we have v ∗ ≤ v + ≤ sup x ∈ R d g ( x ) . Therefore v + ∈ U + . The o ther par t is iden tical.  Remark 2. 5. Using L emma 2.2, one c ould e asily show t hat v + is a visc osity sup er- solution of (2) (i.e. satisfies (8) b elow in the visc osity sense) and v − is a visc osity subsolution of (2) (i.e. satisfies (7) b elow in the visc osity s en se). However, while true, this do es not pr esent much inter est. The follo wing is the main tec hnical r esult of the pre s en t note: Theorem 2.1. (Sto chastic Perron’s Metho d) If g is b ounde d and LSC then v − is a b oun de d and LSC visc osity sup ersolution of (3)  − u t − L t u ≥ 0 , u ( T , x ) ≥ g ( x ) . If g is b ounde d and USC then v + is a b ounde d and USC visc osity subsolution of (4)  − u t − L t u ≤ 0 , u ( T , x ) ≤ g ( x ) . STOCHASTIC PERRON’S METHOD 5 Remark 2.6. We have v − ( T , x ) ≤ g ( x ) and v + ( T , x ) ≥ g ( x ) by c onstruction. Ther efor e, the t erminal c onditions in (3) and (4) c an b e r eplac e d by e qualities. Remark 2 .7. We would like to p oint out that the semi-c ontinuous solutions ob- taine d by Perr on ’s metho d have the c orr e ct semic ontinuity ne e de d for such a defi- nition. We r efer the r e ader to [1] for an intr o duction t o (semic ontinuous) visc osity sub and sup er-solutions of se c ond or der e quations. Pr o of. W e will only pr ov e that v + is a subsolutio n of (4): the other part is sym- metric. Step 1 . The interior sub-solution pr op erty. Note tha t we a lready know that v + is bo unded a nd upp er semicont inu ous. Let ϕ : [0 , T ] × R d → R be a C 1 , 2 -test function function and assume that v + − ϕ attains a str ic t lo cal max - im um (an a ssumption which is not restr ictive) equal to zero at so me interior p oint ( t 0 , x 0 ) ∈ (0 , T ) × R d . Ass ume that v + do es not satisfy the viscos ity subs o lution prop erty , and therefore − ϕ t ( t 0 , x 0 ) − L t ϕ ( t 0 , x 0 ) > 0 . Since the co efficients of the SDE are contin uous, we conclude that there exists a small eno ugh ball B ( t 0 , x 0 , ε ) suc h that − ϕ t − L t ϕ > 0 on B ( t 0 , x 0 , ε ) , and ϕ > v + on B ( t 0 , x 0 , ε ) − ( t 0 , x 0 ) . Since v + − ϕ is upp er-semico n tin tuous and B ( t 0 , x 0 , ε ) − B ( t 0 , x 0 , ε/ 2) is compact, this means that there exist a δ > 0 suc h that ϕ − δ ≥ v + on B ( t 0 , x 0 , ε ) − B ( t 0 , x 0 , ε/ 2) . Now, if w e choo se 0 < η < δ w e ha v e that the function ϕ η = ϕ − η satisfies the prop erties − ϕ η t − L t ϕ η > 0 on B ( t 0 , x 0 , ε ) , ϕ η > v + on B ( t 0 , x 0 , ε ) − B ( t 0 , x 0 , ε/ 2) . and ϕ η ( t 0 , x 0 ) = v + ( t 0 , x 0 ) − η. Now, we define the new function v η =  v + ∧ ϕ η on B ( t 0 , x 0 , ε ) , v + outside B ( t 0 , x 0 , ε ) . W e clearly hav e v η is upp er-s e mico nt inous and v η ( t 0 , x 0 ) = ϕ η ( t 0 , x 0 ) < v + ( t 0 , x 0 ) . Also, v η satisfies the terminal condition (since ε can b e chosen so that T > t 0 + ε and v + satisfies the terminal condition). It only remains to show that v η ∈ U + to obtain a contradiction. F or the analytica l Perron metho d on viscosity solution, the pro of would now be finished, since the viscosity so lution prop erty is lo c al and the minim um of t w o sup erso lutions is a sup ersolution. In our case, the super martingale 6 ERHAN BA YRAKT AR AND MI HAI S ˆ IRBU prop erty defining U + is glob al so we need to lo calize it using stopping times. Partic- ular care has to be taken since the paths may not b e r ight-con tinous, so lo ca lization in gener al may fail, a s p ointed out in Remark 2.3. Fix ( s, x ) and  ( X s,x t ) s ≤ t ≤ T , ( W s,x t ) s ≤ t ≤ T , Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T  ∈ X s,x . W e need to show that the pr o c ess ( v η ( t, X s,x t )) s ≤ t ≤ T is a super martingale on (Ω s,x , P s,x ) with resp ect to the filtration ( F s,x t ) s ≤ t ≤ T . W e first do the pr o of under the a ddi- tional a ssumption that the pr o c ess( v + ( t, X s,x t )) s ≤ t ≤ T do es have RC paths. Under this assumption, the pro cess ( v η ( t, X s,x t )) s ≤ t ≤ T is a sup erma rtingale lo - cally in the region [ s, T ] × R d − B ( t 0 , x 0 , ε/ 2) because it coincides there with the pr o cess ( v + ( t, X s,x t )) s ≤ t ≤ T which is a RC sup erma rtingale so it can b e lo- calized. In addition, in the reg ion B ( t 0 , x 0 , ε ) the pro cess ( v η ( t, X s,x t )) s ≤ t ≤ T is the minimum b et ween tw o lo cal sup erma r tingales, therefore a lo cal s uper martin- gale. (It is clear that the pro cess ( ϕ η ( t, X s,x t )) s ≤ t ≤ T is a lo cal sup ermar tingale over B ( t 0 , x 0 , ε ) b y Itˆ o’s for m ula.) Since the t wo r e gions [ s, T ] × R d − B ( t 0 , x 0 , ε/ 2) and B ( t 0 , x 0 , ε ) actually ov erlap ov er an op en region, then we can conclude that the pro cess ( v η ( t, X s,x t )) s ≤ t ≤ T is indeed a sup ermar tingale. In o r der to make this argument, one needs to cho ose a double s equence of stopping times reminiscent of the optimal s trategy in switching con trol pro blems. Mor e precisely , the double sequence is chosen a s the times ex iting fro m B ( t 0 , x 0 , ε ) and the sequel times en- tering B ( t 0 , x 0 , ε/ 2). The choice depends o n wher e the pro ce ss actually is a time the initial time s . In genera l, i.e., if the pro ces s ( v + ( t, X s,x t )) s ≤ t ≤ T do es not hav e RC paths, then we can w ork with its righ t co n tin uous limit ov e r rational times to reduce it to the case ab ove. Mo re precisely , fix 0 ≤ s ≤ r ≤ t ≤ T and x ∈ R d . W e wan t to pr ov e the sup erma rtingale prop erty for the pro ces s ( Y u ) s ≤ u ≤ T := ( v η ( u, X s,x u )) s ≤ u ≤ T betw een the times r and t , which mea ns we wan t to show that (5) Y r ≥ E s,x [ Y t |F s,x r ] . First, we ma ke the no tation Z u := v + ( u, X s,x u ) for r ≤ u ≤ t and we stop it a t time t , i.e. Z u := v + ( t, X s,x t ) for t ≤ u ≤ T . The pro cess ( Z u ) r ≤ u ≤ T is a sup ermartinga le, but may not b e RC, as discussed. W e can use Prop os ition 3.14 pag e 16 in K aratzas and Shreve [3] to define the RC sup ermar tingale Z + u ( ω ) := lim q → u,q >u,q ∈ Q Z q ( ω ) , ω ∈ Ω ∗ , r ≤ u ≤ T , and Z + · = 0 , ω / ∈ Ω ∗ , where P s,x [Ω ∗ ] = 1. W e would like to emphasize that Z + is, indeed a RC su- per martingale with respect to the original filtr a tion since the filtr a tion is ass umed to satisfy the usual conditions. Since the function v + is USC, and the pr o cess is constant after t we ca n conclude (taking path-wise limits) that Z r ≥ Z + r , Z t = Z + t . W e recall that in the op en region B ( t 0 , x 0 , ε ) − B ( t 0 , x 0 , ε/ 2) w e have v + < ϕ − δ . Therefore, if we take right limits inside this region, and use the fact that ϕ is continous the w e get Z + u < ϕ η ( u, X s,x u ) , if ( u, X s,x u ) ∈ B ( t 0 , x 0 , ε ) − B ( t 0 , x 0 , ε/ 2) . STOCHASTIC PERRON’S METHOD 7 Now, we ca n define the pro ces s Y + u :=  Z + u , ( u, X s,x u ) / ∈ B ( t 0 , x 0 , ε/ 2) , Z + u ∧ ϕ η ( u, x s,x u ) , ( u, X s,x u ) ∈ B ( t 0 , x 0 , ε ) . W e note that we have Y r ≥ Y + r , Y t = Y + t . Now, for the pro cess Y + , we can apply the previous argument, s ince Z + has R C paths, to conclude it is a sup ermar ting ale. In particular, we hav e that Y r ≥ Y + r ≥ E s,x [ Y + t |F s,x r ] = E s,x [ Y t |F s,x r ] . Step 2. The terminal c onditio n . Assume that, for s ome x 0 ∈ R d we have v + ( T , x 0 ) > g ( x 0 ) . W e wan t to use this infor mation in a similar wa y to Step 1 to co ns truct a contradiction. Since g is USC on R d , there ex is ts an ε > 0 such that g ( x ) ≤ v + ( T , x 0 ) − ε, | x − x 0 | ≤ ε. W e now use the fact that v + is USC to c o nclude it is b ounded ab ov e on the compact set ( B ( T , x 0 , ε ) − B ( T , x 0 , ε/ 2)) ∩ ([0 , T ] × R d ) . This was a n ywa y clea r, since a ctually v + is globally b ounded, but the arg umen t ab ov e shows the pr o of works in e v en more genera l cases. Now, we cho ose η > 0 small eno ugh so that (6) v + ( T , x 0 ) + ε 2 4 η ≥ ε + sup ( t,x ) ∈ ( B ( T ,x 0 ,ε ) − B ( T ,x 0 ,ε/ 2)) ∩ ([0 ,T ] × R d ) v + ( t, x ) . W e now define, for k > 0 the fo llowing function ϕ η, ε,k ( t, x ) = v + ( T , x 0 ) + | x − x 0 | 2 η + k ( T − t ) . F or k large enough we hav e that − ϕ ε,η, k t − L t ϕ ε,η, k > 0 , on B ( T , x 0 , ε ) . In addition, using (6) we hav e that ϕ ε,η, k ≥ ε + v + , o n ( B ( T , x 0 , ε ) − B ( T , x 0 , ε/ 2)) ∩ ([0 , T ] × R d ) . Also, ϕ ε,η, k ( T , x ) ≥ v + ( T , x 0 ) ≥ g ( x ) + ε for | x − x 0 | ≤ ε . Now, we can choos e δ < ε and define as in the pro of of Step 1 v ε,η, k ,δ = ( v + ∧  ϕ ε,η, k − δ  on B ( T , x 0 , ε ) , v + outside B ( T , x 0 , ε ) . W e can now prov e, using the sa me switc hing principle and RC modification a r gu- men t as in Step 1 that v ε,η, k ,δ ∈ U + , but v ε,η, k ,δ ( T , x 0 ) = v + ( T , x 0 ) − δ , lea ding to a co ntradiction.  8 ERHAN BA YRAKT AR AND MI HAI S ˆ IRBU 2.2. V erification b y comparison. Definition 2.3. We say that t he visc osity c omp arison principle holds for t he e qua- tion (2) with r esp e ct t o time horizon T and the final c ondi tion g , or that c onditio n C P ( T , g ) is satisfie d if, whenever we have a b oun de d, u pp er-c ontinuous (USC) su b- solution u of (7)  − u t − L t u ≤ 0 , u ( T , x ) ≤ g ( x ) , and a b oun de d lower s emic ontinous sup er-solution v of (8)  − u t − L t u ≥ 0 , u ( T , x ) ≥ g ( x ) . then u ≤ v . Next theore m is a n easy cons equence o f our main result, Theorem 2.1. How ev er, it amount s to a verification result for non- smo oth v iscosity solutions o f (2), so we consider it to be the other main res ult of the prese nt note. Theorem 2.2. L et g b e b ounde d and c ontinous. Assume also that the c omp arison principle C P ( T , g ) is satisfie d. Then ther e exists a unique b ounde d and c ontinuous visc osity solution v t o (2) which e quals b oth t he lower and t he upp er p ay-offs (value functions), which me ans v ∗ = v = v ∗ . In addition, for e ach ( s, x ) ∈ [0 , T ] × R d , and e ach we ak solution  ( X s,x t ) s ≤ t ≤ T , ( W s,x t ) s ≤ t ≤ T , Ω s,x , F s,x , P s,x , ( F s,x t ) s ≤ t ≤ T  ∈ X s,x , the pr o c ess ( v ( t, X s,x )) s ≤ t ≤ T is a mart ingale on (Ω s,x , P s,x ) with r esp e ct to the filtr ation ( F s,x t ) s ≤ t ≤ T . Pr o of. The proo f is immedia te in light of Definition 2 .3, Lemma 2.5 and Theor em 2.1.  Remark 2. 8. The martingale pr op erty for ( v ( t, X s,x )) s ≤ t ≤ T is pr ove d without us- ing the Markov pr op erty of the we ak s olut ion (which is not even assume d). However, if C P ( T , g ) is satisfie d for any T and any b ounde d (test function) g , then we obtai n that, for e ach ( s, x ) and e ach T the law of X s,x T is un iquely determine d. F ol lowing The or em 6.2.3 in [4] or Pr op osition 4.27 p age 326 in [3 ] , uniqueness of mar ginals implies uniqueness in law of t he we ak solut ion. Now, uniqueness in law for any ( s, x ) do es imply the Markov pr op erty for t he we ak solution of t he SDE (1) (t he Markov pr op ert y holds with r esp e ct t o the natura l r aw filt r ation though). We r efer the r e ader to The or em 6.2.2 in [4 ] or The or em 4.20 p age 322 in [3 ] for the last mentione d result. Remark 2.9. The whole p ap er c an b e r ewritten by sele cting, for e ach ( s, x ) , only one we ak solution X s,x inste ad of using the s et of we ak solutions X s,x . S uch a sele ction uses t he axiom of choic e and do es n ot ne e d to b e a Markov sele ction. Onc e the sele ction X s,x is chosen, the sets of sto chastic su p er and sub- solutions in Definitions 2.1 and 2.2 have to b e re -define d ac c or di ngly, and ther e is only one p ay-off (value function) v r eplaci ng v ∗ and v ∗ . STOCHASTIC PERRON’S METHOD 9 3. Conclusions W e desig ned a sto chastic counterpart to Perron’s metho d which pro duces tw o viscosity solutions of the F eynman-Kac equation. The tw o so lutio ns sq ueeze in betw een the exp ected pay-off, and this c o mparison is a trivial c o nsequence of the probabilistic definition. If, in addition, a visco sit y compa rison res ult holds , then we do hav e a unique viscosity solution, which is a ma r tingale along the solutions of the sto chastic differen tial eq uation and is equa l to the exp ected pay-off. In this case, we therefor e have a full verification res ult witho ut smo othness of the visco sity solution. While the Perron metho d we describ e her e is reminiscent of the characteriza tion of the v alue function in optimal stopping pr oblems as the least excessive function, we would like to p o int o ut that here, unlike in optimal stopping, we a v oid proving that the v alue function is “excessive”. This is ac tually the point of verification b y compariso n, to av oid working with the v alue function. One co uld try to prove directly , avoiding viscos it y a lto gether, tha t v + and v − along solutions of the SDE are mar tingales, i.e. they are s to chastic s o lutions in the sense of Stro o ck and V aradhan [5]. How ev er, this is actua lly not p ossible: v + and v − are stochastic solutions only when they coincide, since the sto chastic solutions are unique by definition. W e can additionally justify that, in g eneral, v + and v − are viscosity so lutions but may not b e sto chastic so lutions by obser ving that the sto chastic solution prop erty is muc h stronger then vis c osity . F ortunately , in a larg e nu m be r of situatio ns, viscosity pro pe r t y is still strong enough to prov e uniqueness. In this case, vis cosity and sto chastic s olution pro per t y are equiv alent. 4. Appendix: Count able selection to achieve the inf/sup o f a class of semi-continous functions The main purpo se of the App endix is to prove a countable selec tio n ar gument needed in the pro of of Lemma 2.1. Let ( M , d ) b e a metric spa c e and consider a class G of functions f : M → R . The first res ult is Lemma 4 .1. Le t g : M → R . Then, the fol lowing c onditio ns ar e e qu ivalent (i) g ( x ) = inf f ∈G f ( x ) , for e ach x ∈ M , (ii) { x ∈ M | g ( x ) < q } = ∪ f ∈F { x ∈ M | f ( x ) < q } , for e ach q ∈ Q . Prop ositio n 4.1 . Assume t hat ( M , d ) is a sep ar able metric sp ac e (or, less, a t op o- lo gic al sp ac e with a c ountable b ase). Assume also that e ach function in the class G is upp er-semic ontinous (USC). Then, ther e exist s a c ount able sub class of functions H ⊂ G s u ch that f ∗ ( x ) := inf f ∈G f ( x ) = inf f ∈H f ( x ) , for e ach x ∈ M . Pr o of. Fix a q ∈ Q . According to Lemma 4.1, the op en set { x ∈ M | f ∗ ( x ) < q } admits an op en cover as { x ∈ M | f ∗ ( x ) < q } = ∪ f ∈G { x ∈ M | f ( x ) < q } . Since the space ( M , d ) is separ able, so it admits a countable basis, o ne can select a c ountable op en sub-cover. More precisely , there e xists a c ountable G q ⊂ G such that { x ∈ M | f ∗ ( x ) < q } = ∪ f ∈G q { x ∈ M | f ( x ) < q } . 10 ERHAN BA YRAKT AR AND MI HAI S ˆ IRBU Now, we define the countable class H := ∪ q ∈ Q G q . W e hav e, for each q ∈ Q that { x ∈ M | f ∗ ( x ) < q } = ∪ f ∈G q { x ∈ M | f ( x ) < q } ⊂ ∪ f ∈H { x ∈ M | f ( x ) < q } ⊂ { x ∈ M | f ∗ ( x ) < q } . According to Lemma 4 .1 we then ha v e that f ∗ ( x ) = inf f ∈H f ( x ) , for each x ∈ M .  Remark 4.1 . We would like to p oint out that in the ar gument of sele cting a c ount- able op en sub-c over the axiom of choic e is use d. References 1. M. Crandall, H. Ishii, and P .-L. Lions, User’s g uide to visc osity solutions of se c ond-or der p artial differ e ntial equa tions , Bull. Amer. Math. So c 27 (1992), 1–67. 2. H. Ishii, Perr on ’s metho d for Hamilton-Jac obi e quations , Duk e Mathematical Journal 55 (1987), no. 2, 369–384. 3. I. Karatzas and S. Shrev e, Br ownian motion and st o chastic c alculus , Springer New Y ork, 1988. 4. D. W. Stro ock and S. R . S. V aradhan, Multidimensional diffusion pr o c esses , Class i cs i n Math- ematics, Springer-V erlag, Berl in, 2006, Repri n t of the 1997 edition. 5. D.W. Stro o c k and S.R.S. V aradhan, On de gener ate el liptic - p ar ab olic op er ators of sec ond or der and their asso ciate d diffusions , Comm unications on Pure and Applied Mathematics 25 (1972), 651–713. University of Michigan, Dep ar tmen t of M a them a tics, 530 Church Street, Ann Arbor, MI 48109 . E-mail addr ess : erhan@umich .edu. University of Texas a t Austin, Dep ar tment of Ma thema tics, 1 University St a tion C1200, Austin, TX, 7 8712. E-mail addr ess : sirbu@math. utexas.e du.