Adaptive optimal allocation in stratified sampling methods

A daptiv e optimal allo ation in stratied sampling metho ds Pierre Etoré ∗ , Benjamin Jourdain † Otob er 25, 2018 Abstrat In this pap er, w e prop ose a stratied sampling algorithm in whi h the random dra wings made in the strata to ompute the exp etation of in terest are also used to adaptiv ely mo dify the prop ortion of further dra wings in ea h stratum. These prop ortions on v erge to the optimal allo ation in terms of v ariane redution. And our stratied estimator is asymptotially normal with asymptoti v ariane equal to the minimal one. Numerial exp erimen ts onrm the eieny of our algorithm. In tro dution Let X b e a R d -v alued random v ariable and f : R d → R a measurable funtion su h that E ( f 2 ( X )) < ∞ . W e are in terested in the omputation of c = E ( f ( X )) using a stratied sampling Mon te-Carlo estimator. W e supp ose that ( A i ) 1 ≤ i ≤ I is a partition of R d in to I str ata su h that p i = P [ X ∈ A i ] is kno wn expliitely for i ∈ { 1 , . . . , I } . Up to remo ving some strata, w e assume from no w on that p i is p ositiv e for all i ∈ { 1 , . . . , I } . The stratied Mon te-Carlo estimator of c (see [G04 ℄ p.209-235 and the referenes therein for a presen tation more detailed than the urren t in tro dution) is based on the equalit y E ( f ( X )) = P I i =1 p i E ( f ( X i )) where X i denotes a random v ariable distributed aording to the onditional la w of X giv en X ∈ A i . Indeed, when the v ariables X i are sim ulable, it is p ossible to estimate ea h exp etation in the righ t-hand-side using N i i.i.d dra wings of X i . Let N = P I i =1 N i b e the total n um b er of dra wings (in all the strata) and q i = N i / N denote the prop ortion of dra wings made in stratum i . Then b c is dened b y b c = I X i =1 p i N i N i X j =1 f ( X j i ) = 1 N I X i =1 p i q i q i N X j =1 f ( X j i ) , where for ea h i the X j i 's, 1 ≤ j ≤ N i , are distributed as X i , and all the X j i 's, for 1 ≤ i ≤ I , 1 ≤ j ≤ N i are dra wn indep enden tly . This stratied ∗ CERMICS, Univ ersité P aris Est, 6-8 a v en ue Blaise P asal, Cité Desartes, Champs-sur- Marne, 77455 Marne la V allée Cedex 2, e-mail : etoreermis.enp .fr, supp orted b y the ANR pro jet AD AP'MC † pro jet team Math, CERMICS, Univ ersité P aris Est, 6-8 a v en ue Blaise P asal, Cité Desartes, Champs-sur-Marne, 77455 Marne la V allée Cedex 2, e-mail : jour- dainermis.enp .fr 1 sampling estimator an b e implemen ted for instane when X is distributed aording to the Normal la w on R d , A i = { x ∈ R d : y i − 1 < u ′ x ≤ y i } where −∞ = y 0 < y 1 < . . . < y I − 1 < y I = + ∞ and u ∈ R d is su h that | u | = 1 . Indeed, then one has p i = N ( y i ) − N ( y i − 1 ) with N ( . ) denoting the um ulativ e distribution funtion of the one dimensional normal la w and it is easy to sim ulate aording to the onditional la w of X giv en y i − 1 < u ′ X ≤ y i (see setion 3.2 for a n umerial example in the on text of options priing). W e ha v e E ( b c ) = c and V ( b c ) = I X i =1 p 2 i σ 2 i N i = 1 N I X i =1 p 2 i σ 2 i q i = 1 N I X i =1  p i σ i q i  2 q i ≥ 1 N  I X i =1 p i σ i q i q i  2 , (0.1) where σ 2 i = V ( f ( X i )) = V ( f ( X ) | X ∈ A i ) for all 1 ≤ i ≤ I . During all the sequel w e onsider that ( H ) σ i > 0 for at least one index i. The brute fore Mon te Carlo estimator of E f ( X ) is 1 N P N j =1 f ( X j ) , with the X j 's i.i.d. dra wings of X . Its v ariane is 1 N   I X i =1 p i ( σ 2 i + E 2 ( f ( X i ))) − I X i =1 p i E ( f ( X i )) ! 2   ≥ 1 N I X i =1 p i σ 2 i . F or giv en strata the stratied estimator a hiev es v ariane redution if the allo ations N i or equiv alen tly the prop ortions q i are prop erly  hosen. F or in- stane, for the so-alled prop ortional allo ation q i = p i , ∀ i , the v ariane of the stratied estimator is equal to the previous lo w er b ound of the v ariane of the brute fore Mon te Carlo estimator. F or the  hoie q i = p i σ i P I j =1 p j σ j =: q ∗ i , ∀ 1 ≤ i ≤ I , the lo w er-b ound in ( 0.1 ) is attained. W e sp eak of optimal al lo  ation . W e then ha v e V ( b c ) = 1 N  I X i =1 p i σ i  2 =: σ 2 ∗ N , and no  hoie of the q i 's an a hiev e a smaller v ariane of b c . In general when the onditional exp etations E ( f ( X ) | X ∈ A i ) = E ( f ( X i )) are unkno wn, then so are the onditional v ariane σ 2 i . Therefore optimal al- lo ation of the dra wings is not feasible at one. One an of ourse estimate the onditional v arianes and the optimal prop ortions b y a rst Mon te Carlo algorithm and run a seond Mon te Carlo pro edure with dra wings indep enden t from the rst one to ompute the stratied estimator orresp onding to these estimated prop ortions. But, as suggested in [A04 ℄ in the dieren t on text of imp ortane sampling metho ds, it is a pit y not to use the dra wings made in the rst Mon te Carlo pro edure also for the nal omputation of the onditional exp etations. Instead of running t w o suessiv e Mon te Carlo pro edures, w e an think to get a rst estimation of the σ i 's, using the rst dra wings of the X i 's made to 2 ompute the stratied estimator. W e ould then estimate the optimal allo a- tions b efore making further dra wings allo ated in the strata aording to these estimated prop ortions. W e an next get another estimation of the σ i 's, om- pute again the allo ations and so on. Our goal is th us to design and study su h an adaptive str atie d estimator . The estimator is desrib ed in Setion 1 . In partiular, w e prop ose a v ersion of the algorithm su h that at ea h step, the allo ation of the new dra wings in the strata is not simply prop ortional to the urren t estimation of the optimal prop ortions but  hosen in order to minimize the v ariane of the stratied estimator at the end of the step. A Cen tral Limit Theorem for this estimator is sho wn in Setion 2 . The asymptoti v ariane is equal to the optimal v ariane σ 2 ∗ and our estimator is asymptotially optimal. In Setion 3 , w e onrm the eieny of our algorithm b y n umerial exp erimen ts. W e rst deal with a to y example b efore onsidering the priing of an arithmeti a v erage Asian option in the Bla k-S holes mo del. Another stratied sampling algorithm in whi h the optimal prop ortions and the onditional exp etations are estimated using the same dra wings has b een v ery reen tly prop osed in [CGL07 ℄ for quan tile estimation. More preisely , for a total n um b er of dra wings equal to N , the authors suggest to allo ate the N γ with 0 < γ < 1 rst ones prop ortionally to the probabilities of the strata and then use the estimation of the optimal prop ortions obtained from these rst dra wings to allo ate the N − N γ remaining ones. Their stratied estimator is also asymptotially normal with asymptoti v ariane equal to the optimal one. In pratie, N is nite and it is b etter to tak e adv an tage of all the dra wings and not only the N γ rst ones to mo dify adaptiv ely the allo ation b et w een the strata. Our algorithm w orks in this spirit. 1 The algorithm The onstrution of the adaptiv e stratied estimator relies on steps at whi h w e estimate the onditional v arianes and ompute the allo ations. W e denote b y N k the total n um b er of dra wings made in all the strata up to the end of step k . By on v en tion, w e set N 0 = 0 . In order to b e able to mak e one dra wing in ea h stratum at ea h step w e assume that N k − N k − 1 ≥ I for all k ≥ 1 . F or all 1 ≤ i ≤ I w e denote b y N k i the n um b er of dra wings in stratum i till the end of step k with on v en tion N 0 i = 0 . The inremen ts M k i = N k i − N k − 1 i 's are omputed at the b eginning of step k using the information on tained in the N k − 1 rst dra wings. STEP k ≥ 1 . Computation of the empiri al varian es. If k > 1 , for all 1 ≤ i ≤ I ompute b σ k − 1 i = v u u u t 1 N k − 1 i  N k − 1 i X j =1 ( f ( X j i )) 2 −  1 N k − 1 i N k − 1 i X j =1 f ( X j i )  2  . If k = 1 , set b σ 0 i = 1 for 1 ≤ i ≤ I . Computation of the al lo  ations M k i = N k i − N k − 1 i . 3 W e mak e at least one dra wing in ea h stratum. This ensures the on v ergene of the estimator and of the b σ k i 's (see the pro of of Prop osition 1.1 b elo w). That is to sa y w e ha v e, ∀ 1 ≤ i ≤ I , M k i = 1 + ˜ m k i , with ˜ m k i ∈ N , (1.1) and w e no w seek the ˜ m k i 's. W e ha v e P I i =1 ˜ m k i = N k − N k − 1 − I , and p ossibly ˜ m k i = 0 for some indexes. W e presen t t w o p ossible w a ys to ompute the ˜ m k i 's. a) W e kno w that the optimal prop ortion of total dra wings in stratum i for the stratied estimator is q ∗ i = p i σ i P I j =1 p j σ j , so w e ma y w an t to  ho ose the v etor ( ˜ m k 1 , . . . , ˜ m k I ) ∈ N I lose to ( m k 1 , . . . , m k I ) ∈ R I + dened b y m k i = p i b σ k − 1 i P I j =1 p j b σ k − 1 j ( N k − N k − 1 − I ) for 1 ≤ i ≤ I . This an b e a hiev ed b y setting ˜ m k i = ⌊ m k 1 + . . . + m k i ⌋ − ⌊ m k 1 + . . . + m k i − 1 ⌋ , with the on v en tion that the seond term is zero for i = 1 . This systemati sampling pro edure ensures that P I i =1 ˜ m k i = N k − N k − 1 − I and m k i − 1 < ˜ m k i < m k i + 1 for all 1 ≤ i ≤ I . In ase b σ k − 1 i = 0 for all 1 ≤ i ≤ I , the ab o v e denition of m k i do es not mak e sense and w e set m k i = p i ( N k − N k − 1 − I ) for 1 ≤ i ≤ I b efore applying the systemati sampling pro edure. Note that thanks to ( H ) and the on v ergene of the b σ k i (see Prop osition 1.1 b elo w), this asymptotially will nev er b e the ase. b) In ase b σ k − 1 i = 0 for all 1 ≤ i ≤ I , w e do as b efore. Otherwise, w e ma y think to the expression of the v ariane of the stratied estimator with allo ation N i for all i , whi h is giv en b y (0.1), and nd ( m k 1 , . . . , m k I ) ∈ R I + that minimizes I X i =1 p 2 i ( b σ k − 1 i ) 2 N k − 1 i + 1 + m k i , under the onstrain t P I i =1 m k i = N k − N k − 1 − I . This an b e done in the follo wing manner (see in the App endix Prop osi- tion 4.1 ): F or the indexes i su h that b σ k − 1 i = 0 , w e set m k i = 0 . W e denote I k the n um b er of indexes su h that b σ k − 1 i > 0 . W e ren um b er the orresp onding strata from 1 to I k . W e no w nd ( m k 1 , . . . , m k I k ) ∈ R I k + that minimizes P I k i =1 p 2 i ( b σ k − 1 i ) 2 N k − 1 i +1+ m k i , under the onstrain t P I k i =1 m k i = N k − N k − 1 − I , b y applying the three follo wing p oin ts: i) Compute the quan tities N k − 1 i +1 p i b σ k − 1 i and sort them in dereasing order. Denote b y N k − 1 ( i ) +1 p ( i ) b σ k − 1 ( i ) the ordered quan tities. 4 ii) F or i = 1 , . . . , I k ompute the quan tities N k − N k − 1 − I + I k X j = i +1 ( N k − 1 ( j ) + 1) I k X j = i +1 p ( j ) b σ k − 1 ( j ) . Denote b y i ∗ the last i su h that N k − 1 ( i ) + 1 p ( i ) b σ k − 1 ( i ) ≥ N k − N k − 1 − I + I k X j = i +1 ( N k − 1 ( j ) + 1) I k X j = i +1 p ( j ) b σ k − 1 ( j ) . If this inequalit y is false for all i , then b y on v en tion i ∗ = 0 . iii) Then for i ≤ i ∗ set m k ( i ) = 0 and for i > i ∗ , m k ( i ) = p ( i ) b σ k − 1 ( i ) . N k − N k − 1 − I + I k X j = i ∗ +1 ( N k − 1 ( j ) + 1) I k X j = i ∗ +1 p ( j ) b σ k − 1 ( j ) − N k − 1 ( i ) − 1 . This quan tit y is non-negativ e aording to the pro of of Prop osition 4.1 . W e then build ( m k 1 , . . . , m k I ) b y reinluding the I − I k zero v alued m k i 's and using the initial indexation. Finally w e dedue ( ˜ m k 1 , . . . , ˜ m k I ) ∈ N I b y the systemati sampling pro edure desrib ed in a) . Dr awings of the X i 's. Dra w M k i i.i.d. realizations of X i in ea h stratum i and set N k i = N k − 1 i + M k i . Computation of the estimator Compute ˆ c k := I X i =1 p i N k i N k i X j =1 f ( X j i ) . (1.2) Square in tegrabilit y of f ( X ) is not neessary in order to ensure that the estimator b c k is strongly onsisten t. Indeed thanks to ( 1.1), w e ha v e N k i → ∞ as k → ∞ and the strong la w of large n um b ers ensures the follo wing Prop osition. Prop osition 1.1 If E | f ( X ) | < + ∞ , then b c k − − − − → k →∞ c a . s .. If mor e over, E ( f 2 ( X )) < + ∞ , then a.s., ∀ 1 ≤ i ≤ I , b σ k i − − − − → k →∞ σ i and I X i =1 p i b σ k i − − − − → k →∞ σ ∗ . 5 2 Rate of on v ergene In this setion w e pro v e the follo wing result. Theorem 2.1 Assume ( H ) , E ( f 2 ( X )) < + ∞ and k / N k → 0 as k → ∞ . Then, using either pr o  e dur e a) or pr o  e dur e b) for the  omputation of al lo  ations, one has √ N k  ˆ c k − c  inlaw − − − − → k →∞ N (0 , σ 2 ∗ ) . With Prop osition 1.1 , one dedues that √ N k P I i =1 p i b σ k i  ˆ c k − c  inlaw − − − − → k →∞ N (0 , 1 ) , whi h enables the easy onstrution of ondene in terv als. The theorem is a diret onsequene of the t w o follo wing prop ositions. Prop osition 2.1 If E ( f 2 ( X )) < + ∞ and ∀ 1 ≤ i ≤ I , N k i N k − − − − → k →∞ q ∗ i a . s ., (2.1) then √ N k  ˆ c k − c  inlaw − − − − → k →∞ N (0 , σ 2 ∗ ) . Prop osition 2.2 Under the assumptions of The or em 2.1 , using either pr o  e- dur e a) or pr o  e dur e b) for the  omputation of al lo  ations, ( 2.1) holds. W e pro v e Prop osition 2.1 and 2.2 in the follo wing subsetions. 2.1 Pro of of Prop osition 2.1 The main to ol of the pro of of this prop osition will b e a CL T for martingales that w e reall b elo w. Theorem 2.2 (Cen tral Limit Theorem) L et ( µ n ) n ∈ N b e a squar e-inte gr able ( F n ) n ∈ N -ve tor martingale. Supp ose that for a deterministi se quen e ( γ n ) in- r e asing to + ∞ we have, i) h µ i n γ n P − − − − → n →∞ Γ . ii) The Lindeb er g  ondition is satise d, i.e. for al l ε > 0 1 γ n n X k =1 E h || µ k − µ k − 1 || 2 1 {|| µ k − µ k − 1 ||≥ ε √ γ n } |F k − 1 i P − − − − → n →∞ 0 . Then µ n √ γ n inlaw − − − − → n →∞ N (0 , Γ) . 6 As w e an write √ N k  ˆ c k − c  =     p 1 N k N k 1 . . . p I N k N k I     . 1 √ N k     P N k 1 j =1 ( f ( X j 1 ) − E f ( X 1 )) . . . P N k I j =1 ( f ( X j I ) − E f ( X I ))     , w e ould think to set µ k :=  P N k 1 j =1 ( f ( X j 1 ) − E f ( X 1 )) , . . . , P N k I j =1 ( f ( X j I ) − E f ( X I ))  ′ and try to use Theorem 2.2 . Indeed if w e dene the ltration ( G k ) k ∈ N b y G k = σ ( 1 j ≤ N k i X j i , 1 ≤ i ≤ I , 1 ≤ j ) , it an b e sho wn that ( µ k ) is a ( G k ) - martingale. This is thanks to the fat that the N k i 's are G k − 1 -measurable. Then easy omputations sho w that 1 N k h µ i k = diag   N k 1 N k σ 2 1 , . . . , N k I N k σ 2 I   where diag( v ) denotes the diagonal matrix with v etor v on the diagonal. Thanks to (2.1) w e th us ha v e 1 N k h µ i k a . s . − − − − → k →∞ diag   q ∗ 1 σ 2 1 , . . . , q ∗ I σ 2 I   , and a use of Theorem 2.2 and Slutsky's theorem ould lead to the desired result. The trouble is that Lindeb erg's ondition annot b e v eried in this on text, and w e will not b e able to apply Theorem 2.2 . Indeed the quan tit y || µ k − µ k − 1 || 2 in v olv es N k − N k − 1 random v ariables of the t yp e X i and w e annot on trol it without making some gro wth assumption on N k − N k − 1 . In order to handle the problem, w e are going to in tro due a mirosopi sale. F rom the sequene of estimators (ˆ c k ) w e will build a sequene (˜ c n ) of estimators of c , su h that ˆ c k = ˜ c N k , and for whi h w e will sho w a CL T using Theorem 2.2 . It will b e p ossible b eause it in v olv es a new martingale ( µ n ) su h that µ n − µ n − 1 is equal to a v etor the only non zero o ordinate of whi h is one random v ariable f ( X j i ) . Then the Lindeb erg ondition will b e easily v eried, but this time w e will ha v e to w ork a little more to  he k the bra k et ondition. As the sequene (ˆ c k ) is a subsequene of (˜ c n ) , Prop osition 2.1 will follo w. This is done in the follo wing w a y . Let n ∈ N ∗ . In the setting of the Algorithm of Setion 1 let k ∈ N su h that N k − 1 < n ≤ N k . Giv en the allo ations ( N l i ) I i =1 , for 0 ≤ l ≤ k , w e dene for ea h 1 ≤ i ≤ I a quan tit y ν n i with the indutiv e rule b elo w. Ea h ν n i is the n um b er of dra wings in the i -th strata among the rst n dra wings and w e ha v e P I i =1 ν n i = n . W e then dene e c n := I X i =1 p i ν n i ν n i X j =1 f ( X j i ) . 7 Rule for the ν n i 's F or n = 0 , ν n i = 0 , for all 1 ≤ i ≤ I . 1. F or k > 0 set r k i := N k i − N k − 1 i N k − N k − 1 for 1 ≤ i ≤ I . 2. F or N k − 1 < n ≤ N k , and giv en the ν n − 1 i 's nd i n = argma x 1 ≤ i ≤ I  r k i − ν n − 1 i − N k − 1 i n − N k − 1  . If sev eral i realize the maxim um  ho ose i n to b e the one for whi h r k i is the greatest. If there are still ex aequo's  ho ose the greatest i . 3. Set ν n i n = ν n − 1 i n + 1 , and ν n i = ν n − 1 i if i 6 = i n . There is alw a ys an index i for whi h r k i − ν n − 1 i − N k − 1 i n − N k − 1 > 0 , sine I X i =1 ν n − 1 i − N k − 1 i n − N k − 1 = n − 1 − N k − 1 n − N k − 1 < 1 = I X i =1 r k i . Moreo v er, for the rst n ∈ { N k − 1 + 1 , . . . , N k } su h that ν n − 1 i = N k i in the i -th strata, r k i − ν n − 1 i − N k − 1 i n − N k − 1 ≤ 0 and ν n ′ i = ν n i = N k i for n ≤ n ′ ≤ N k . This implies that ν N k i = N k i , ∀ 1 ≤ i ≤ I , ∀ k ∈ N , and as a onsequene, ˆ c k = ˜ c N k . (2.2) Therefore Prop osition 2.1 is an easy onsequene of the follo wing one. Prop osition 2.3 Under the assumptions of Pr op osition 2.1 , √ n  ˜ c n − c  inlaw − − − − → n →∞ N (0 , σ 2 ∗ ) . In the pro of of Prop osition 2.3 , to v erify the bra k et ondition of Theorem 2.2 , w e will need the follo wing result. Lemma 2.1 When (2.1) holds, then ∀ 1 ≤ i ≤ I , ν n i n − − − − → n →∞ q ∗ i a . s . Pro of. Let b e 1 ≤ i ≤ I . During the sequel, for x ∈ R ∗ + or n ∈ N ∗ , the in teger k is impliitely su h that N k − 1 < x, n ≤ N k . W e notie that for an y n ∈ N ∗ ν n i n = n − N k − 1 n . ν n i − N k − 1 i n − N k − 1 + N k − 1 n . N k − 1 i N k − 1 , 8 and dene for x ∈ R ∗ + , f ( x ) := x − N k − 1 x . N k i − N k − 1 i N k − N k − 1 + N k − 1 x . N k − 1 i N k − 1 . W e will see that, as n tends to innit y , f ( n ) tends to q ∗ i and f ( n ) − ν n i n tends to zero. Computing the deriv ativ e of f on an y in terv al ( N k − 1 , N k ] w e nd that this funtion is monotoni on it. Besides f ( N k − 1 ) = N k − 1 i N k − 1 and f ( N k ) = N k i N k . So if N k i N k tends to q ∗ i as k tends to innit y , w e an onlude that f ( n ) − − − − → n →∞ q ∗ i . (2.3) As r k i = N k i − N k − 1 i N k − N k − 1 w e no w write ν n i n − f ( n ) = n − N k − 1 n  ν n i − N k − 1 i n − N k − 1 − r k i  . W e onlude the pro of b y  he king that r k i − I − 1 n − N k − 1 < ν n i − N k − 1 i n − N k − 1 < r k i + 1 n − N k − 1 . (2.4) Indeed, this inequalit y implies − I − 1 n < ν n i n − f ( n ) < 1 n , whi h om bined with (2.3 ) giv es the desired onlusion. W e rst sho w ν n i − N k − 1 i n − N k − 1 < r k i + 1 n − N k − 1 . (2.5) W e distinguish t w o ases. Either ν n ′ i = N k − 1 i for all N k − 1 < n ′ ≤ n , that is to sa y no dra wing at all is made in stratum i b et w een N k − 1 and n , then (2.5) is trivially v eried. Either some dra wing is made b et w een N k − 1 and n . Let us denote b y n ′ the index of the last one, i.e. w e ha v e ν n i = ν n ′ i = ν n ′ − 1 i + 1 . As a dra wing is made at n ′ w e ha v e ν n ′ − 1 i − N k − 1 i n ′ − N k − 1 < r k i . W e th us ha v e, ν n ′ − 1 i − N k − 1 i n − N k − 1 ≤ ν n ′ − 1 i − N k − 1 i n ′ − N k − 1 < r k i and ν n i − N k − 1 i n − N k − 1 = ν n ′ − 1 i + 1 − N k − 1 i n − N k − 1 , and th us w e ha v e again (2.5 ) . Using no w the fat that 1 = P I i =1 r k i = P I i =1 ν n i − N k − 1 i n − N k − 1 w e get ν n i − N k − 1 i n − N k − 1 = r k i + X i 6 = j  r k j − ν n j − N k − 1 i n − N k − 1  Using this and (2.5 ) w e get ( 2.4 ) . 9 Pro of of Prop osition 2.3 . F or n ≥ N 1 , ν n i ≥ 1 for all 1 ≤ i ≤ I and w e an write √ n  ˜ c n − c  =     p 1 n ν n 1 . . . p I n ν n I     . 1 √ n µ n , (2.6) with µ n =     P ν n 1 j =1 ( f ( X j 1 ) − E f ( X 1 )) . . . P ν n I j =1 ( f ( X j I ) − E f ( X I ))     . Note that if σ i = 0 for a stratum i , then q ∗ i = 0 and b y Lemma 2.1 , n ν n i a . s . − − − − → n →∞ + ∞ whi h ma y ause some trouble in the on v ergene analysis. In omp ensa- tion, σ i = 0 means that f ( X i ) − E f ( X i ) = 0 a.s. Th us the omp onen t µ i n of µ n mak es no on tribution in ˜ c n − c . So w e migh t rewrite ( 2.6 ) with µ n a v etor of size less than I , whose omp onen ts orresp ond only to indexes i with σ i > 0 . F or the seek of simpliit y w e k eep the size I and onsider that σ i > 0 for all 1 ≤ i ≤ I . If w e dene F n := σ ( 1 j ≤ ν n i X j i , 1 ≤ i ≤ I , 1 ≤ j ) , then ( µ n ) n ≥ 0 is ob viously a ( F n ) -martingale. Indeed, for n ∈ N ∗ let k ∈ N ∗ su h that N k − 1 < n ≤ N k . F or 1 ≤ i ≤ I the v ariables N k − 1 i and N k i are resp etiv ely F N k − 2 and F N k − 1 - measurable (Step k > 1 in the Algorithm). As for ea h 1 ≤ i ≤ I the quan tit y ν n i dep ends on the N k − 1 i 's and the N k i 's, it is F N k − 1 -measurable. Th us µ n is F n -measurable and easy omputations sho w that E [ µ n +1 |F n ] = µ n . W e wish to use Theorem 2.2 with γ n = n . W e will denote b y diag( a i ) the I × I matrix ha ving n ull o eien ts exept the i -th diagonal term with v alue a i . W e rst v erify the Lindeb erg ondition. W e ha v e, using the sequene ( i n ) dened in the rule for the ν n i 's, 1 n P n l =1 E  || µ l − µ l − 1 || 2 1 {|| µ l − µ l − 1 || >ε √ n } |F l − 1  = 1 n P n l =1 E  | f ( X ν l i l i l ) − E f ( X i l ) | 2 1 {| f ( X ν l i l i l ) − E f ( X i l ) | >ε √ n } |F l − 1  ≤ 1 n P n l =1 sup 1 ≤ i ≤ I E  | f ( X i ) − E f ( X i ) | 2 1 {| f ( X i ) − E f ( X i ) | >ε √ n }  = sup 1 ≤ i ≤ I E  | f ( X i ) − E f ( X i ) | 2 1 {| f ( X i ) − E f ( X i ) | >ε √ n }  . As sup 1 ≤ i ≤ I E  | f ( X i ) − E f ( X i ) | 2 1 {| f ( X i ) − E f ( X i ) | >ε √ n }  − − − − → n →∞ 0 , the Lindeb erg ondition is pro v en. W e no w turn to the bra k et ondition. W e ha v e, h µ i n = P n k =1 E  ( µ k − µ k − 1 )( µ k − µ k − 1 ) ′ |F k − 1  = P n k =1 diag  E    f ( X ν k i k i k ) − E f ( X i k )   2   = P n k =1 diag  σ 2 i k  . 10 Th us, w e ha v e h µ i n n = diag  ( ν n 1 n σ 2 1 , . . . , ν n I n σ 2 I )  − − − − → n →∞ diag  ( q ∗ 1 σ 2 1 , . . . , q ∗ I σ 2 I )  a . s ., where w e ha v e used Lemma 2.1 . Theorem 2.2 implies that µ n √ n inlaw − − − − → n →∞ N  0 , diag  ( q ∗ 1 σ 2 1 , . . . , q ∗ I σ 2 I )   . (2.7) Using again Lemma 2.1 w e ha v e ( p 1 n ν n 1 , . . . , p I n ν n I ) − − − − → n →∞ ( p 1 q ∗ 1 , . . . , p I q ∗ I ) a . s . (2.8) Using nally Slutsky's theorem, (2.6 ), (2.7 ) and (2.8 ), w e get, √ n  ˜ c n − c  inlaw − − − − → n →∞ N  0 , σ 2 ∗ ) . 2.2 Pro of of Prop osition 2.2 Thanks to ( H ) and Prop osition 1.1 there exists K ∈ N s.t. for all k ≥ K w e ha v e P I i =1 p i b σ k i > 0 . The prop ortions ( ρ k i = p i b σ k i P I j =1 p j b σ k j ) i are w ell dened for all k ≥ K and pla y an imp ortan t role in b oth allo ation rules a) and b) . Prop osition 1.1 implies on v ergene of ρ k i as k → + ∞ . Lemma 2.2 Under the assumptions of The or em 2.1 , ∀ 1 ≤ i ≤ I , ρ k i − − − − → k →∞ q ∗ i a . s . Pro of of Prop osition 2.2 for allo ation rule a) . Let b e 1 ≤ i ≤ I . W e ha v e N k i N k = k + P k l =1 ˜ m l i N k . Using the fat that m l i − 1 < ˜ m l i < m l i + 1 w e an write P k l =1 m l i N k ≤ N k i N k ≤ 2 k N k + P k l =1 m l i N k . W e will sho w that P k l =1 m l i N k → q ∗ i , and, as k N k → 0 , will get the desired result. F or k ≥ K + 1 , w e ha v e P k l =1 m l i N k = P K l =1 m l i N k + P k l = K +1 ρ l i ( N l − N l − 1 − I ) N k = P K l =1 m l i N k + N k − N K N k × 1 N k − N K N k X n = N K +1 ˜ ρ n i − I ( k − K ) N k × 1 k − K k X l = K ρ l i where the sequene ( ˜ ρ n i ) dened b y ˜ ρ n i = ρ l i for N l − 1 < n ≤ N l on v erges to q ∗ i as n tends to innit y . The Cesaro means whi h app ear as fators in the seond and third terms of the r.h.s. b oth on v erge a.s. to q ∗ i . One easily dedue that the rst, seond and third terms resp etiv ely on v erge to 0 , q ∗ i and 0 . 11 Pro of of Prop osition 2.2 for allo ation rule b) . There ma y b e some strata of zero v ariane. W e denote b y I ′ ( I ′ ≤ I ) the n um b er of strata of non zero v ariane. F or a stratum i of zero v ariane the only dra wing made at ea h step will b e the one fored b y ( 1.1 ) . Indeed b σ k i = 0 for all k in this ase. Th us N k i = k for all the strata of zero v ariane and sine k N k → 0 , w e get the desired result for them (note that of ourse q ∗ i = 0 in this ase). W e no w w ork on the I ′ strata su h that σ i > 0 . W e ren um b er these strata from 1 to I ′ . Let no w K ′ b e su h that b σ k i > 0 for all k ≥ K ′ , and all 1 ≤ i ≤ I ′ . F or k ≥ K ′ , the in teger I k +1 at step k + 1 in pro edure b) is equal to I ′ . Step 1. W e will rstly sho w that ∀ k ≥ K ′ , ∀ 1 ≤ i ≤ I ′ N k +1 i N k +1 ≤ N k i + 1 N k +1 ∨  ρ k i + 1 N k +1  . (2.9) Let k ≥ K ′ . A t step k + 1 w e denote b y ( . ) k the ordered index in P oin t i) of pro edure b) and b y i ∗ k the index i ∗ in P oin t ii). W e also set n k +1 i = N k i + 1 + m k +1 i . By P oin t iii), for i > i ∗ k , n k +1 ( i ) k p ( i ) k b σ k ( i ) k = m k +1 ( i ) k + N k ( i ) k + 1 p ( i ) k b σ k ( i ) k = N k +1 − N k − I + P I ′ j = i ∗ k +1 ( N k ( j ) k + 1) P I ′ j = i ∗ k +1 p ( j ) k b σ k ( j ) k (2.10) Case 1: i ∗ k = 0 . Then, in addition to the dra wing fored b y (1.1 ), there are some dra wings at step k + 1 in stratum (1) k , and onsequen tly in all the strata. Th us (2.10 ) leads to n k +1 i = ρ k i   N k +1 − N k − I + I ′ + I ′ X j =1 N k j   , ∀ 1 ≤ i ≤ I ′ . But N k = P I ′ j =1 N k j + k ( I − I ′ ) and, follo wing the systemati sampling pro edure, w e ha v e N k +1 i < n k +1 i + 1 , ∀ 1 ≤ i ≤ I ′ . (2.11) Th us, in this ase, N k +1 i N k +1 ≤ ρ k i + 1 N k +1 , ∀ 1 ≤ i ≤ I ′ . Case 2: i ∗ k > 0 . If i ≤ i ∗ k , N k +1 ( i ) k = N k ( i ) k + 1 and (2.9 ) holds. If i > i ∗ k , then (2.10 ) leads to n k +1 ( i ) k N k +1 = ρ k ( i ) k N k +1 − N k − I + P I ′ j = i ∗ k +1 ( N k ( j ) k + 1) N k +1 P I ′ j = i ∗ k +1 ρ k ( j ) k . Using (2.11 ) , it is enough to  he k that N k +1 − N k − I + P I ′ j = i ∗ k +1 ( N k ( j ) k + 1) N k +1 P I ′ j = i ∗ k +1 ρ k ( j ) k ≤ 1 (2.12) 12 in order to dedue that ( 2.9 ) also holds for i > i ∗ k . If N k ( i ∗ k ) k +1 N k +1 ρ k ( i ∗ k ) k ≤ 1 , then inequalit y (2.12 ) holds b y the denition of i ∗ k . If N k ( i ∗ k ) k +1 N k +1 ρ k ( i ∗ k ) k > 1 w e ha v e N k ( i ) k +1 N k +1 ρ k ( i ) k > 1 , ∀ i ≤ i ∗ k and th us i ∗ k X j =1 ( N k ( j ) k + 1) > N k +1 i ∗ k X j =1 ρ k ( j ) k . This inequalit y also writes N k − k ( I − I ′ ) + I ′ − I ′ X j = i ∗ k +1 ( N k ( j ) k + 1) > N k +1  1 − I ′ X j = i ∗ k +1 ρ k ( j ) k  , and (2.12 ) follo ws. Step 2. Let 1 ≤ i ≤ I ′ . W e set ¯ n k i := N k i − k (this the n um b er of dra wings in stratum i that ha v e not b een fored b y (1.1)). Using (2.9 ) w e ha v e ∀ k ≥ K ′ , N k +1 i − ( k + 1 ) N k +1 ≤ N k i + 1 − ( k + 1) N k +1 ∨  ρ k i − k N k +1  , and th us ∀ k ≥ K ′ , ¯ n k +1 i N k +1 ≤ ¯ n k i N k +1 ∨  ρ k i − k N k +1  . Let ε > 0 . Thanks to Lemma 2.2 , there exists k 0 ≥ K ′ s.t. for all k ≥ k 0 , ρ k i − k N k +1 ≤ q ∗ i + ε . Th us ∀ k ≥ k 0 , ¯ n k +1 i N k +1 ≤ ¯ n k i N k +1 ∨  q ∗ i + ε ) . (2.13) By indution ∀ k ≥ k 0 , ¯ n k i N k ≤ ¯ n k 0 i N k ∨ ( q ∗ i + ε ) . Indeed supp ose ¯ n k i N k ≤ ¯ n k 0 i N k ∨ ( q ∗ i + ε ) . If ¯ n k i N k ≤ q ∗ i + ε then ¯ n k i N k +1 ≤ q ∗ i + ε and using (2.13 ) w e get ¯ n k +1 i N k +1 ≤ q ∗ i + ε . Otherwise ¯ n k i = ¯ n k 0 i and using (2.13 ) w e are done. But as ¯ n k 0 i N k → 0 as k → ∞ w e dedue that lim sup k ¯ n k i N k ≤ q ∗ i + ε . Sine this is true for an y ε , and k N k → 0 , w e an onlude that lim sup k N k i N k ≤ q ∗ i . No w using the indexation on all the strata and the result for the strata with v ariane zero, w e dedue that for 1 ≤ i ≤ I , lim inf k N k i N k = lim inf k  1 − P I j =1 j 6 = i N k j N k  ≥ 1 − P I j =1 j 6 = i lim sup k N k j N k = 1 − P I j =1 j 6 = i q ∗ j = q ∗ i . This onludes the pro of. 13 3 Numerial examples and appliations to option priing 3.1 A rst simple example W e ompute c = E X where X ∼ N (0 , 1) . Let I = 10 . W e  ho ose the strata to b e giv en b y the α -quan tiles y α of the normal la w for α = i/I for 1 ≤ i ≤ I . That is to sa y A i = ( y i − 1 I , y i I ] for all 1 ≤ i ≤ I , with the on v en tion that y 0 = −∞ and y 1 = + ∞ . In this setting w e ha v e p i = 1 / 10 for all 1 ≤ i ≤ I . Let us denote b y d ( x ) the densit y of the la w N (0 , 1 ) . Thanks to the relation d ′ ( x ) = − xd ( x ) and using in tegration b y parts, w e an establish that, for all 1 ≤ i ≤ I , E  X 1 y i − 1 I 0 b e the option's maturit y and  t m = mT d  1 ≤ m ≤ d the sequene of times when the v alue of the underlying asset is monitored to ompute the a v erage. The disoun ted pa y o of the arithmeti Asian option with strik e K is giv en b y e − r T  1 d d X m =1 S t m − K  + . 16 Th us the prie of the option is giv en b y c = E h e − r T  1 d d X m =1 S t m − K  + i . But in this Bla k-S holes setting w e an exatly sim ulate the S t m 's using the fat that S t 0 = S 0 and S t m = S t m − 1 exp  [ r − 1 2 V 2 ]( t m − t m − 1 ) + V p t m − t m − 1 X m  , ∀ 1 ≤ m ≤ d, (3.1) where X 1 , . . . , X d are indep enden t standard normals. Th us, c = E [ g ( X ) 1 D ( X )] , with g some deterministi funtion, D = { x ∈ R d : g ( x ) > 0 } , and X a R d - v alued random v ariable with la w N (0 , I d ) . In [GHS99 ℄ the authors disuss and link together t w o issues: imp ortane sampling and stratied sampling. Their imp ortane sampling te hnique onsists in a  hange of mean of the gaussian v etor X . Let us denote b y h ( x ) the densit y of the la w N (0 , I d ) and b y h µ ( x ) the densit y of the la w N ( µ, I d ) for an y µ ∈ R d . W e ha v e, c = Z D g ( x ) h ( x ) h µ ( x ) h µ ( x ) dx = E [ g ( X + µ ) h ( X + µ ) h µ ( X + µ ) 1 D ( X + µ )] . The v ariane of g ( X + µ ) h ( X + µ ) h µ ( X + µ ) 1 D ( X + µ ) is giv en b y Z D  g ( x ) h ( x ) h µ ( x ) − c  2 h µ ( x ) dx. Heuristially , this indiates that an eetiv e  hoie of h µ should giv e w eigh t to p oin ts for whi h the pro dut of the pa y o and the densit y is large. In other w ords, if w e dene G ( x ) = log g ( x ) w e should lo ok for µ ∈ R that v eries, µ = argmax x ∈ D  G ( x ) − 1 2 x ′ x  (3.2) The most signian t part of the pap er [GHS99℄ is aimed at giving an asymp- totial sense to this heuristi, using large deviations to ols. The idea is then to sample g ( X + µ ) h ( X + µ ) h µ ( X + µ ) 1 D ( X + µ ) . Standard omputations sho w that for an y µ ∈ R d , c = E  g ( X + µ ) e − µ ′ X − (1 / 2) µ ′ µ 1 D ( X + µ )  . Th us the problem is no w to build a Mon te Carlo estimator of c = E f µ ( X ) , sam- pling f µ ( X ) with X ∼ N (0 , I d ) , and with f µ ( x ) = g ( x + µ ) e − µ ′ x − (1 / 2) µ ′ µ 1 D ( x + µ ) , for the v etor µ satisfying ( 3.2 ) . The authors of [ GHS99 ℄ then prop ose to use a stratied estimator of c = E f µ ( X ) . Indeed for u ∈ R d with u ′ u = 1 , and a < b real n um b ers, it is easy to sample aording to the onditional la w of X giv en u ′ X ∈ [ a, b ] . 17 It an b e done in the follo wing w a y (see Subsetion 4.1 of [ GHS99℄ for de- tails). W e rst sample Z = Φ − 1 ( V ) with Φ − 1 the in v erse of the um ulativ e normal distibution, and V = Φ( a ) + U (Φ( b ) − Φ( a )) , with U uniform on [0 , 1] . Seond w e sample Y ∼ N (0 , I d ) indep enden t of Z . W e then ompute, X = uZ + Y − u ( u ′ Y ) , whi h b y on trution has the desired onditional la w. Let b e u ∈ R d satisfy u ′ u = 1 . With our notation the stratied estimator b c in [GHS99℄ is built in the follo wing w a y . They tak e I = 10 0 . As in subsetion 3.1 w e denote b y y α the α -quan tile of the la w N (0 , 1 ) . F or all 1 ≤ i ≤ I , they tak e A i = { x ∈ R d : y i − 1 I < u ′ x ≤ y i I } . That is to sa y X i has the onditional la w of X giv en y i − 1 I < u ′ X ≤ y i I , for all 1 ≤ i ≤ I . As in this setting u ′ X ∼ N (0 , 1) , they ha v e p i = 1 /I for all 1 ≤ i ≤ I . They then do prop ortional allo ation, that is to sa y , N i = p i N for all 1 ≤ i ≤ I , where N is the total n um b er of dra wings (in other w ords q i = p i ). Then, the v ariane of their stratied estimator is 1 N I X i =1 p i σ 2 i . A ording to the In tro dution, that  hoie ensures v ariane redution. The question of the  hoie of the pro jetion diretion u arises. The authors tak e u = µ/ ( µ ′ µ ) , with the v etor µ satisfying (3.2 ) that has b een used for the imp ortane sampling. They laim that this pro vides in pratie a v ery eien t pro jetion diretion, for their stratied estimator with prop ortional allo ation. As  P I i =1 p i σ i  2 ≤ P I i =1 p i σ 2 i (i.e. prop ortional allo ation is sub optimal), if u is a go o d pro jetion diretion for a stratied estimator with prop ortional allo ation, it is a go o d diretion for a stratied estimator with optimal allo a- tion. In the sequel w e tak e the same diretion u and the same strata as in [GHS99 ℄, and disuss allo ation. Indeed w e ma y wish to do optimal allo ation and tak e q i = q ∗ i = p i σ i P j p j σ j . The trouble is the analytial omputation of the quan tities σ 2 i = V ( f µ ( X ) | u ′ X ∈ ( y i − 1 I , y i I ]) , is not tratable, at least when f µ is not linear. As the p i 's are kno wn, this is exatly the kind of situation where our adaptiv e stratied estimator an b e useful. 3.2.2 The results In all the tests w e ha v e tak en S 0 = 50 , V = 0 . 1 , r = 0 . 05 and T = 1 . 0 . The total n um b er of dra wings is N = 100 0000 . W e all GHS the pro edure used in [GHS99℄, that is imp ortane sampling plus stratied sampling with prop ortional allo ation. W e all SSAA our pro- edure, that is the same imp ortane sampling plus stratied sampling with adaptiv e allo ation. 18 d K Prie v ariane SSAA ratio GHS/SSAA 16 45 6.05 2 . 37 × 1 0 − 8 2.04 50 1.91 1 . 00 × 1 0 − 7 35 55 0.20 5 . 33 × 1 0 − 9 39.36 64 45 6.00 3 . 36 × 1 0 − 9 3.34 50 1.84 9 . 00 × 1 0 − 10 1.60 55 0.17 6 . 40 × 1 0 − 9 61 T able 1: Results for a all option with S 0 = 50 , V = 0 . 1 , r = 0 . 05 , T = 1 . 0 and N = 10000 00 (and I = 100 ). More preisely in the pro edure SSAA w e  ho ose N 1 = 10 0000 , N 2 = 40000 0 , N 3 = 500000 and ompute our adaptiv e stratied estimator ˆ c 3 of c = E f ( X ) , with the same strata as in GHS. W e ha v e used pro edure a) for the omputation of allo ations. W e denote b y ¯ c the GHS estimator of c . W e all v ariane GHS or v ariane SSAA the quan tit y b σ , whi h is an estimation of the v ariane of ¯ c or ˆ c 3 . More preisely for GHS, ( b σ ) 2 = 1 N I X i =1 p i b σ i 2 , where for ea h 1 ≤ i ≤ I , b σ i 2 = 1 p i N p i N X j =1 f 2 ( X j i ) −  1 p i N p i N X j =1 f ( X j i )  2 , and for SSAA ( b σ ) 2 = 1 N  I X i =1 p i b σ i  2 , where for ea h 1 ≤ i ≤ I , ( b σ i ) 2 = 1 N 3 i N 3 i X j =1 f 2 ( X j i ) −  1 N 3 i N 3 i X j =1 f ( X j i )  2 . T ables 1 and 2 sho w the results resp etiv ely for a all option and a put option. W e all ratio GHS/SSAA the v ariane GHS divided b y the v ariane SSAA. In general the impro v emen t is m u h b etter for a put option. Indeed the v ariane is often divided b y 100 in this ase. A further analysis an explain these results. W e plot on Figure 3 and 4 the v alues of the b σ i 's and the estimated v alues of the onditional exp etations 19 d K Prie v ariane SSAA ratio GHS/SSAA 16 45 0.013 7 . 29 × 1 0 − 10 107 50 0.63 7 . 29 × 1 0 − 8 79 55 3.74 2 . 50 × 1 0 − 5 249 64 45 0.011 5 . 76 × 1 0 − 10 95 50 0.62 5 . 61 × 1 0 − 8 64 55 3.69 1 . 85 × 1 0 − 5 58 T able 2: Results for a put option with S 0 = 50 , V = 0 . 1 , r = 0 . 05 , T = 1 . 0 and N = 10000 00 (and I = 100 ). 0 10 20 30 40 50 60 70 80 90 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 Figure 3: On the left: v alue of b σ i in funtion of the stratum index i in the ase of a all option. On the righ t: estimated v alue of E f µ ( X i ) . (P arameters are the same as in T ables 1 , with d = 64 and K = 45 ). 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 Figure 4: On the left: v alue of b σ i in funtion of the stratum index i in the ase of a put option. On the righ t: estimated v alue of E f µ ( X i ) . (Same parameters than in Figure 3). 20 E f µ ( X i ) 's, for a all and a put option, with d = 64 and K = 45 , a ase for whi h the ratio GHS/SSAA is 3.34 in the all ase and 95 in the put ase. W e observ e that in the ase of the put option the estimated onditional v ariane of ab out 90% of the strata is zero, unlik e in the ase of the all option. These estimated onditional v arianes are zero, b eause in the orresp onding strata the estimated onditional exp etations are onstan t with v alue zero. But these strata are of non zero probabilit y (remem b er that in this setting p i = 0 . 01 , for all 1 ≤ i ≤ 100 ). Th us the GHS pro edure with prop ortional allo ation will in v est dra wings in these strata, resulting in a loss of auray , while in our SSAA pro edure most of the dra wings are made in the strata of non zero estimated v ariane. One an w onder if the exp etation in the strata of zero observ ed exp etation is really zero, or if it is just a n umerial eet. W e dene the deterministi funtion s : R d → R b y s ( x ) = S 0 d d X m =1 exp  m X p =1  [ r − V 2 2 ] T d + V r T d x p   , ∀ x = ( x 1 , . . . , x d ) ′ ∈ R d . With the previous notations, in the put option ase, w e ha v e f µ ( X i ) = 0 a.s., and th us E f µ ( X i ) = 0 , if s ( X i + µ ) ≥ K a.s. (note that i denotes here the stratum index and not the omp onen t of the random v etor X i ). Th us the problem is to study , in funtion of z ∈ R , the deterministi v alues of s ( x + µ ) for x ∈ R d satisfying u ′ x = z . The follo wing fats an b e sho wn. Whatev er the v alue of u or z the quan tit y s ( x + µ ) has no upp er b ound. Th us in the all option ase no onditional exp etation E f µ ( X i ) will b e zero. T o study the problem of the lo w er b ound w e denote b y M the matrix of size d × d giv en b y M =       1 0 . . . 0 1 1 . . . . . . . . . . . . 0 1 . . . . . . 1       , with in v erse M − 1 =       1 0 . . . 0 − 1 1 . . . . . . . . . . . . . . . 0 0 . . . − 1 1       , and b y 1 the d -sized v etor (1 , . . . , 1) ′ . If w e use the  hange of v ariable y = M  [ r − V 2 2 ] T d 1 + V r T d ( x + µ )  , w e an see that minimizing s ( x + µ ) for x ∈ R d satisfying u ′ x = z is equiv alen t to minimizing S 0 d P d m =1 exp( y m ) for y ∈ R d satisfying w ′ y = v , (3.3) where, w = ( M − 1 ) ′ u, and v = u ′  [ r − V 2 2 ] T d 1 + V r T d ( x + µ )  = V r T d ( z + u ′ µ ) + ( r − V 2 2 ) d X m =1 u m . 21 0 10 20 30 40 50 60 70 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Figure 5: V alue of the omp onen t u m of u ∈ R d in funtion of m . If all the omp onen ts of w are strily p ositiv e the lo w er b ound of s ( x + µ ) under the onstrain t u ′ x = z is s ∗ = S 0 d × exp  v − P d m =1 w m log w m P d m =1 w m  × d X m =1 w m . (3.4) If all the omp onen ts of w are strily negativ e w e get the same kind of result b y a  hange of sign. Otherwise the lo w er b ound is zero: it is p ossible to let the y m 's tend to −∞ with (3.3 ) satised. In the n umerial example that w e are analysing the diretion v etor u is the same in the all or put option ases, and its omp onen ts are strily p ositiv e and dereasing with the index (see Figure 5). Th us the omp onen ts of w are stritly p ositiv e and the lo w er b ound is giv en b y s ∗ dened b y ( 3.4). With z taking v alues in the 90 last strata w e ha v e s ∗ > 45 . Th us the onditional exp etations E f µ ( X i ) are truly zero in these strata. W e an then w onder if it is w orth stratifying the part of the real line orre- sp onding to these strata, in other w ords stratifying R d and not only D . Ma yb e stratifying D and making prop ortional allo ation will pro vide a suien t v ari- ane redution. But this w ould require a rst analysis, while our SSAA pro e- dure a v oids automatially to mak e a large n um b er of dra wings in D c . T o onlude on the eieny of our algorithm in this example let us notie that the omputation times of the GHS and SSAA pro edures are nearly the same (less than 1% additional time for the SSAA pro edure). Indeed, unlik e in the to y example of Subsetion 3.1 , the omputation time of the allo ation of the dra wings in the strata is almost negligible in omparison to the other alulations (dra wings et...). 4 App endix W e justify the use of pro edure b) in the follo wing prop osition. 22 Prop osition 4.1 When b σ k − 1 i > 0 for some 1 ≤ i ≤ I , by  omputing at Step k the m k i 's with the pr o  e dur e b) desrib e d in Se tion 1, we nd ( m k 1 , . . . , m k I ) ∈ R I + that minimizes I X i =1 p 2 i ( b σ k − 1 i ) 2 N k − 1 i + 1 + m k i , under the  onstr aint P I i =1 m k i = N k − N k − 1 − I . Pro of. First note that if b σ k − 1 i = 0 for some index i it is lear that w e ha v e to set m k i = 0 and to rewrite the minimization problem for the indexes orresp onding to b σ k − 1 i > 0 . This orresp onds to the v ery b eginning of pro edure b) . F or the seek of simpliit y , and without loss of generalit y , w e onsider in the sequel that b σ k − 1 i > 0 for all 1 ≤ i ≤ I , and th us w ork with the indexation { 1 , . . . , I } . W e will note M = N k − N k − 1 − I , and, for all 1 ≤ i ≤ I , n i = N k − 1 i + 1 , α i = p i b σ k − 1 i , and m i = m k i . W e th us seek ( m 1 , . . . , m I ) ∈ R I + that minimizes P I i =1 α 2 i n i + m i under the onstrain t P I i =1 m i = M . Step 1: L agr angian  omputations. W e write the Lagrangian orresp onding to our minimization problem, for all ( m, λ ) ∈ R I + × R : L ( m, λ ) = I X i =1 α 2 i n i + m i + λ ( I X i =1 m i − M ) = I X i =1 h i ( m i , λ ) − λM . with h i ( x, λ ) =  α 2 i n i + x + λx  for all i . W e rst minimize L ( m, λ ) with resp et to m for a xed λ . F or an y λ ∈ R let us denote m ( λ ) := a rgmin m ∈ R I + L ( m, λ ) . Minimizing L ( m, λ ) with resp et to m is equiv alen t to minimizing h i ( m i , λ ) with resp et to m i for all i . The deriv ativ e of ea h h i ( ., λ ) has the same sign as − α 2 i + λ ( n i + x ) 2 . If λ ≤ 0 w e ha v e m ( λ ) = ( ∞ , . . . ∞ ) . If λ > 0 there are t w o ases to onsider for ea h h i : either λ > α 2 i n 2 i and m i ( λ ) = 0 , or λ ≤ α 2 i n 2 i and m i ( λ ) = p α 2 i /λ − n i . (4.1) T o sum up w e ha v e L ( m ( λ ) , λ ) =                −∞ if λ < 0 , 0 if λ = 0 , P I i =1 h 1 { λ> α 2 i n 2 i } α 2 i n i + 1 { λ ≤ α 2 i n 2 i } (2 α i √ λ − n i λ ) i − M λ if λ > 0 . W e no w lo ok for λ ∗ that maximizes L ( m ( λ ) , λ ) . F or all λ ∈ (0 , ∞ ) w e ha v e, 23 ∂ λ L ( m ( λ ) , λ ) = I X i =1 1 { λ ≤ α 2 i n 2 i }  α i √ λ − n i  − M . (4.2) This funtion is on tin uous on (0 , + ∞ ) , equal to − M for λ ≥ ma x i α 2 i n 2 i , de- reasing on (0 , max i α 2 i n 2 i ] and tends to + ∞ as λ tends to 0 . W e dedue that λ 7→ L ( m ( λ ) , λ ) rea hes its unique maxim um at some λ ∗ ∈ (0 , max i α 2 i n 2 i ) . If ∂ λ L  m  α 2 ( i ) n 2 ( i )  , α 2 ( i ) n 2 ( i )  < 0 for all 1 ≤ i ≤ I , w e set i ∗ = 0 . Otherwise w e sort in inreasing order the α 2 i /n 2 i 's, index with ( i ) the ordered quan tities, and note i ∗ the in teger su h that ∂ λ L  m  α 2 ( i ∗ ) n 2 ( i ∗ )  , α 2 ( i ∗ ) n 2 ( i ∗ )  ≥ 0 and ∂ λ L  m  α 2 ( i ∗ +1) n 2 ( i ∗ +1)  , α 2 ( i ∗ +1) n 2 ( i ∗ +1)  < 0 . (4.3) Then λ ∗ b elongs to  α 2 ( i ∗ ) n 2 ( i ∗ ) , α 2 ( i ∗ +1) n 2 ( i ∗ +1)  , or  0 , α 2 (1) n 2 (1)  if i ∗ = 0 . But on this in terv al ∂ λ L ( m ( λ ) , λ ) = I X j = i ∗ +1 ( α ( j ) √ λ − n ( j ) ) − M . As ∂ λ L ( m ( λ ∗ ) , λ ∗ ) = 0 w e ha v e, 1 √ λ ∗ = M + I X j = i ∗ +1 n ( j ) I X j = i ∗ +1 α ( j ) . Clearly , if i ∗ 6 = 0 , λ ∗ ≥ α 2 ( i ) n 2 ( i ) is equiv alen t to i ≤ i ∗ . If i ∗ = 0 then λ ∗ < α 2 ( i ) n 2 ( i ) for all 1 ≤ i ≤ I . Th us, aording to (4.1), w e ha v e m ( i ) ( λ ∗ ) = 0 if i ≤ i ∗ , and if i > i ∗ , m ( i ) ( λ ∗ ) = α ( i ) . M + I X j = i ∗ +1 n ( j ) I X j = i ∗ +1 α ( j ) − n ( i ) . (4.4) W e ha v e th us found ( m ( λ ∗ ) , λ ∗ ) that satises L ( m ( λ ∗ ) , λ ∗ ) = max λ ∈ R min m ∈ R I + L ( m, λ ) , whi h implies that L ( m ( λ ∗ ) , λ ∗ ) ≤ L ( m, λ ∗ ) for all m ∈ R I + . Besides (4.4 ) implies P I i =1 m i ( λ ∗ ) = M and L ( m ( λ ∗ ) , λ ∗ ) = L ( m ( λ ∗ ) , λ ) for all λ ∈ R . Therefore ( m ( λ ∗ ) , λ ∗ ) is a saddle p oin t of the Lagrangian and m ( λ ∗ ) solv es the onstrained minimization problem. 24 Step 2. W e no w lo ok for a riterion to nd the index i ∗ satifying (4.3 ). If i ∗ 6 = 0 , w e ha v e the follo wing equiv alenes using the ona vit y of λ 7→ L ( m ( λ ) , λ ) and (4.2) i ≤ i ∗ ⇔ ∂ λ L ( m ( α 2 ( i ) n 2 ( i ) ) , α 2 ( i ) n 2 ( i ) ) ≥ 0 ⇔ n ( i ) α ( i ) ≥ M + I X j = i +1 n ( j ) I X j = i +1 α ( j ) . In the same manner, i ∗ = 0 ⇔ n ( i ) α ( i ) < M + I X j = i +1 n ( j ) I X j = i +1 α ( j ) , ∀ 1 ≤ i ≤ I . The pro of of Prop osition 4.1 in then ompleted: in P oin ts i) and ii) of pro edure b) w e nd the index i ∗ men tionned in Step 1, using the riterion of Step 2. In P oin t iii) w e ompute the solution of the optimization problem using the results of Step 1. Referenes [A04℄ B. Arouna. A daptative Monte Carlo metho d, a varian e r e dution te hnique . Mon te Carlo Metho ds Appl. V ol. 10, No. 1 (2004), 1-24. [CGL07℄ C. Cannamela, J. Garnier and B. Lo oss. Contr ol le d str ati ation for quantile estimation . Preprin t (2007), submitted to Annals of Applied Statistis. [G04℄ P . Glasserman. Monte Carlo metho ds in nanial engine ering . Springer V erlag (2004). [GHS99℄ P . Glasserman, P . Heidelb erger and P . Shahabuddin. Asymptoti Optimal Imp ortan e Sampling and Str ati ation for Priing Path- Dep endent Options . Mathematial Finane, V ol. 9, No. 2 (1999), 117-152. 25