Simulation of BSDEs and. Wiener Chaos Expansions

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1 Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp:// Workshop on BSDEs Rennes, May 22-24, 213

2 Inroducion Simulaion of BSDEs and Wiener Chaos Expansions Inroducion Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

3 Inroducion Backward Sochasic Differenial Equaions A BSDE is an equaion of he ype T T Y = ξ + f (s, Y s, Z s )ds Z s db s, T The firs resul was proved by É. Pardoux and S. Peng Theorem (Pardoux-Peng) If he generaor f is Lipschiz, and [ ] E ξ 2 + T f (s,, ) 2 ds < + hen he BSDE has a unique square inegrable soluion Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

4 Inroducion Euler s scheme for BSDEs For a SDE For a BSDE X +h = X + h b(x ) + σ(x ) (B +h B ) Y h = Y + h f (Y h, Z h ) Z h (B B h ) The correc equaion is Y h = Y + h f (Y h, Z h ) Z h (B B h ) + (N N h ) The soluion is given by Z h = h 1 E (Y (B B h ) F h ), Y h = E (Y F h ) + h f (Y h, Z h ) Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

5 Inroducion Discreizaion of BSDEs No so easy as for SDEs D. Chevance 1997, V. Bally 1997 Dynamical Programming Equaion A "sepwise consan" Brownian moion in his case (Ph. B., B. Delyon, J. Mémin 1 & 2) The speed of convergence was proved by J. Zhang 4 and B. Bouchard and N. Touzi 4 for Markovian BSDEs Error expansion by E. Gobe and C. Labar 7 L 2 regulariy E. Gobe, A. Makhlouf 1, Quadraic case A. Richou 11 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

6 Inroducion Discreizaion of BSDEs One of he main difficulies is o compue condiional expecaions Several approaches : using Malliavin calculus, B. Bouchard and N. Touzi using regression mehods, E. Gobe, J.-P. Lemor and X. Warin using quanizaion, V. Bally and G. Pagès We propose here o use Wiener chaos expansions In he spiri, i is no so far from he regression echnics C. Bender and R. Denk 7 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

7 Inroducion How o solve a BSDE? Two seps : The generaor does no depend on (Y, Z ) Fixed poin argumen Picard ieraions ξ, {f (s)} s T square inegrable T T Y = ξ + f (s) ds Z s db s, T. The soluion is given by Y = E ( ξ + T f (s) ds ) F Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

8 Inroducion How o solve a BSDE? T Seing F = ξ + f (s) ds, we have Y = E (F F ) f (s) ds Finally, he soluion is given by T F = E [F ] + Z s db s. Y = E (F F ) f (s) ds Z = D Y = D E (F F ) Explici formulae using Wiener chaos expansions + Picard ieraions Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

9 The Algorihm Simulaion of BSDEs and Wiener Chaos Expansions The Algorihm Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

10 The Algorihm Wiener Chaos Expansion Muliple Wiener Inegrals (Ω, F, P) complee probabiliy space B real valued Brownian moion {F } T augmened filraion of B : F = F T If F L 2 (Ω, F, P), T F =E[F ] u 1 (s 1 )db s1 + T sn T s2 u n is a deerminisic funcion defined on u 2 (s 2, s 1 )db s1 db s2 s2 u n (s n,..., s 1 )db s1... db sn +... S n (T ) := {(s 1,, s n ) [, T ] n : < s 1 <... < s n < T } Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

11 The Algorihm Wiener Chaos Expansion Muliple Wiener Inegrals If u n L 2 (S n (T )), we se J n (u n ) := T sn s2 u n (s n,, s 1 )db s1 db sn, H n := vec ( J n (u n ) : u n L 2 (S n (T )) ) We have E [ J n (u n ) 2] = sn s2 sn E [J n (u n )J m (u m )] = δ nm u 2 n(s n,..., s 1 )ds 1... ds n s2 u n (s n,..., s 1 )u m (s n,..., s 1 )ds 1... ds n Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

12 The Algorihm Wiener Chaos Expansion Muliple Wiener Inegrals To summarize, wih H = R, L 2 (Ω, F, P) = n H n Noaions : for F L 2 (Ω, F, P) P n (F) = J n (u n ) is he projecion of F on H n, C p (F) denoes he chaos decomposiion of F up o order p p p C p (F) = E[F] + P k (F) = E[F ] + J k (u k ) = E[F] + k=1 T u 1 (s 1 )db s k=1 T s2 u p (s p,..., s 1 )db s1... db sp E [ F 2] = E [F ] 2 + n 1 E [ P n (F ) 2] Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

13 The Algorihm Wiener Chaos Expansion Hermie polynomials The Hermie polynomials are defined by he expansion e x 2 /2 = H n (x) n n! = K n (x) n n n H 2 (x) = x 2 1, K 1 (x) = 1 ( x 2 1 ). 2 ( n! ) Kn is an orhonormal basis of L 2 (R, µ) n µ normal disribuion N (, 1) Le (g n ) n 1 be an orhonormal basis of L 2 (, T ) H n is he closure of he vecor field spanned by ( ) T ni! K ni g i (s)db s : (n i ) i 1 := n i = n i 1 i 1 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

14 The Algorihm Wiener Chaos Expansion Hermie polynomials Le Λ be he se of all sequences of inegers (n i ) i 1 such ha n = i 1 n i < + The se ( ) T ni! K ni g i (s)db s : n = (n i ) i 1 Λ i 1 is an orhonormal basis of L 2 (Ω, F, P) For F L 2 (Ω, F, P), F = E[F] + k 1 n =k d n k ( ) T K ni g i (s)db s i 1 dk n = n! E F ( T K ni i 1 g i (s)db s ), n! = i 1 n i! Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

15 The Algorihm Generaor independen of (Y, Z ) We consider he BSDE Y = ξ + A simple BSDE T The soluion is given by T f (s) ds Z s db s, T T Y = E (F F ) f (s) ds, Z = D E (F F ), F = ξ + f (s) ds Compue Y and Z wih he decomposiion F = E[F] + k 1 n =k d n k ( ) T K ni g i (s)db s i 1 dk n = n! E F ( T K ni i 1 g i (s)db s ), n! = i 1 n i! Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

16 The Algorihm Generaor independen of (Y, Z ) Choice of he basis We choose an orhonormal basis of L 2 (, T ) for which compuaions are easy The firs N funcions, g 1,..., g N, are given by g i () = ( i i 1 ) 1/2 1 ]i 1, i ](), i = i T N = ih, i = 1,..., n We complee hese funcions ino an orhonormal basis of L 2 (, T ), (g i ) i 1 We can consider he Haar basis on each inerval ( i 1, i ) Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

17 The Algorihm Generaor independen of (Y, Z ) Approximaion of F We consider a finie numbers of chaos p p F C p (F) = E [F ] + dk n k=1 n =k i 1 We work only wih he funcions g 1,..., g N p F C p (F ) Cp N (F ) = E [F] + k=1 n =k p = E [F] + ( ) T K ni g i (s)db s d n k dk n k=1 n =k i=1 ( n ) T K ni g i (s)db s i=1 n K ni (G i ) dk n = n 1!... n N! E [F K n1 (G 1 )... K nn (G N )] wih he noaions n = (n 1,..., n N ), G i = ( i i 1 ) 1/2 ( ) B i B i 1 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

18 The Algorihm Generaor independen of (Y, Z ) Approximaion of F Roughly speaking, we consider ha H k is generaed by Exemple For p = 2, C N 2 (F) = E [F ] + {K n1 (G 1 )... K nn (G N ) : n n N = k} N j=1 d e j 1 K 1(G j ) + j 1 N d e ij 2 K 1(G i )K 1 (G j ) + j=1 i=1 where (e j ) 1 j N is he canonical basis of R N, and e ij = e i + e j. For j = 1,, N and i = 1,, j 1 d e j 1 = E(FK 1(G j )), d e ij 2 = E(FK 1(G i )K 1 (G j )), d 2e j 2 = 2E(FK 2 (G j )). N j=1 d 2e j 2 K 2(G j ), Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

19 The Algorihm Generaor independen of (Y, Z ) Approximaion of (Y, Z ): coninuous ime For 1 r N and r 1 < r, wih n(r) = (n 1,..., n r ) E ( C N p (F) ) = E [F ] + Y p,n p k=1 n(r) =k d n k ( ) ( ) nr K r 1 2 B B r 1 n i<r i (G i ) Knr, h r 1 ( = E C N p F ) r 1 h f ( i 1 ) ( r 1 )f ( r 1 ), Z p,n = D E ( C N p (F ) ) = h 1/2 p k=1 n(r) =k n r > d n k i=1 ( ) nr 1 ( ) K r 1 2 B B n i<r i (G i ) K r 1 nr 1. h r 1 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

20 The Algorihm Generaor independen of (Y, Z ) Approximaion of (Y, Z ): coninuous ime Y p,n is coninuous on [, T ] To define Z p,n we se Case p = 2, for 1 r N, Y 2,N r = E[F] + Z 2,N r = h 1/2 ( r j=1 Y p,n = E[F ] = d Z p,n = lim Z p,n = d e 1 1 = E[F B h]. h h d e j 1 K 1(G j ) + j 1 r d e ij 2 K 1(G i )K 1 (G j ) + j=1 i=1 d er 1 + d 2er 2 K 1 (G r ) + r 1 i=1 d e ir 2 K 1(G i ) ). r j=1 d 2e j 2 K 2(G j ), Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

21 The Algorihm Generaor independen of (Y, Z ) The Algorihm Saring from M rajecories of he BM B, we ge M rajecories of F We compue d = Ê [F ] = 1 M M m=1 F m d n k = n!ê [FK n 1 (G 1 )... K nn (G N )] = n! 1 M We ge M rajecories of (Y, Z ) Y p,n,m r = d + Z p,n,m r = h 1/2 p k=1 n(r) =k p k=1 n(r) =k n r > d n k d n k M F m K n1 (G1 m )... K nn (GN m ) m=1 i r K n i (G m i ) h r f m ( i 1 ), i=1 i<r K n i (G m i ) K nr 1 (G m r ). Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

22 The Algorihm The general case A BSDE wih a generaor We consider now he BSDE Y = ξ + T f (s, Y s, Z s )ds T Z s db s, T. We assume ha f : [, T ] R R d R is Lipschiz f (, y, z) f (, y, z ) L f ( y y + z z ). The algorihm consiss in wo seps The Picard scheme o remove he generaor The previous approximaion Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

23 The Algorihm The general case Picard Ieraions (Y, Z ) = (, ) and for q T Y q+1 = ξ + f ( s, Ys q, Zs q ) T ds Zs q+1 db s, T. Le ( Y q,p,n, Z q,p,n) be he approximaion of (Y q, Z q ) ( Y,p,N, Z,p,N) = (, ) F q,p,n = ξ + T f (s, Y q,p,n s, Zs q,p,n )ds Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

24 The Algorihm The general case Previous Approximaion We consruc, wih he explici formulas, for r =,..., N Y q+1,p,n r Z q+1,p,n r = E ( Cp N ( F q,p,n ) ) r ( F r h f = D r Y q+1,p,n r i=1 i 1, Y q,p,n i 1 = D r E ( Cp N ( F q,p,n ) ) F r Saring wih M rajecories of B he coefficiens of he Wiener expansions are compued by Mone-Carlo we obain M rajecories of he process ( Y q,p,n,m, Z q,p,n,m) ), Z q,p,n i 1 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

25 Numerical Examples Simulaion of BSDEs and Wiener Chaos Expansions Numerical Examples Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

26 Numerical Examples Linear Driver - Financial Benchmark Linear Driver - Financial Benchmark Black-Scholes model in dimension d = 1 Linear generaor f (, y, z) = ry S = S e (r 1 2 σ2 )+σw, [, T ]. We consider a Discree Down and Ou Barrier Call opion The parameers are ξ = (S T K ) + 1 n [,N]Sn L r =.1, σ =.2, T = 1, K =.9, L =.85, S = 1, N = 2 By a variance reducion Mone Carlo mehod Y = , δ = Z σs =.8327 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

27 Numerical Examples Linear Driver - Financial Benchmark Linear Driver - Financial Benchmark Evoluion of Y_ Evoluion of dela_ 134e e e e 3.64 ref y_app log(m) ref dela_app log(m) Figure : Evoluion of Y q,p,n,m and δ := Z q,p,n,m w.r.. log(m) when N = 2, p = 2, σs q = 5 -Discree Down and Ou Barrier Call opion Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

28 Numerical Examples Non linear driver - Financial Benchmark Non linear driver - Financial Benchmark Black-Scholes model in dimension d = 5 i = 1,, 5 S i = S i e (µi (σi ) 2 2 )+σ i B i. The correlaion marix of B is given by B i, B j = C i,j = ρ1 i j + 1 i=j, 1 4 < ρ < 1 The borrowing rae R is higher han he bond one r The driver is given by f (, y, z) = ry θ z + (R r)(y where θ = θ := Σ 1 (µ r1), Σ ij = σ i L ij, C = LL 5 (Σ 1 z) i ) We consider a Pu Baske Opion : ξ = (K i=1 Si T ) + i=1 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

29 Numerical Examples Non linear driver - Financial Benchmark Non linear driver - Financial Benchmark Comparison wih resuls from É. Gobe & C. Labar 21 CPU ime = 166 when M = 5 and N = 2 Evoluion of Y_ Evoluion of dela_(1) 2.1 ref y_app 2. 45e 3 49e e e e log(m) ref dela_app 65e log(m) Figure : Evoluion of Y q,p,n,m and δ (1) w.r.. log(m) when N = 2, p = 2, q = 5, d = 5 - Baske Pu opion wih differen ineres and borrowing raes. Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

30 Numerical Examples Non linear driver - Financial Benchmark Non linear driver - Financial Benchmark We ry his example in dimension d = 1 The price and he δ are obained in 5 minues Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

31 Numerical Examples Non Linear Driver & Pah Dependen Terminal Condiion Non Linear Driver & Pah Dependen ξ The example is he following : d = 1 ξ = sup T B f (, y, z) = cos(y) Comparison beween p = 2 and p = 3 for N = 2 and M = 1 5 Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

32 Numerical Examples Non Linear Driver & Pah Dependen Terminal Condiion Non Linear Driver & Pah Dependen ξ ieraions CPU ime p = p = Table : Evoluion of Y q,p,n,m w.r.. Picard s ieraions, M = 1 5, N = 2 and CPU ime ieraions CPU ime p = p = Table : Evoluion of Z q,p,n,m w.r.. Picard s ieraions, M = 1 5, N = 2 and CPU ime Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

33 Numerical Examples Non Linear Driver & Pah Dependen Terminal Condiion + + Non Linear Driver & Pah Dependen ξ Convergence in M Evoluion of Y^q_ Evoluion of Z^q_ M=1^3 M=1^4 M=1^5.9 + M=1^3 M=1^4 M=1^ ieraions ieraions Figure : Evoluion of Y q,p,n,m and Z q,p,n,m w.r.. q and M when N = 2, p = 2, ξ = sup T B s, f (, y, z) = cos(y). Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

34 Numerical Examples Non Linear Driver & Pah Dependen Terminal Condiion Non Linear Driver & Pah Dependen ξ Convergence in N Evoluion of Y^q_ Evoluion of Z^q_ N=1 N=2 N=3 N= N=1 N=2 N=3.9 N= ieraions ieraions Figure : Evoluion of Y q,p,n,m and Z q,p,n,m w.r.. N when M = 1 5, p = 2 - ξ = sup T B s, f (, y, z) = cos(y) Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

35 Numerical Examples Non Linear Driver & Pah Dependen Terminal Condiion Conclusion The complexiy of he algorihm is he following O ( K i M p (N d) p+1) This algorihm is raher fas even when he dimension of he underlying BM is grea easy o implemen Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

36 Error Analysis Simulaion of BSDEs and Wiener Chaos Expansions Error Analysis Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

37 Error Analysis Convergence Resuls Spliing he error We wan o conrol he error [ E = E Y Y q,p,n sup T 2 + T Of course, we spli he error ino hree pars E = E q + E q,p + E q,p,n ] Z Z q,p,n 2 d Y Y q,p,n = Y Y q + Y q Y q,p + Y q,p Y q,p,n A erm is missing : Y q,p,n Y q,p,n,m which comes from "Mone Carlo expecaions" The firs par is very easy : E q C(T, L f ) 2 q, C(T, L f ) is explici Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

38 Error Analysis Convergence Resuls A lo of regulariy The generaor f smooh : f C b ξ is smooh in he Malliavin sense : ξ D Hölder coninuiy : E[D (r) s 1,,s r ξ] E[D (r) 1,, r ξ] K ξ ( s 1 1 α ξ + + s r r α ξ ) The Malliavin derivaives belongs o L p Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

39 Error Analysis Convergence Resuls Example X soluion o he SDE X = x + wih b and σ C wih bounded derivaives g : R n R C p b(x s ) ds + σ(x s ) db s, Regulariy assumpion rue ξ = g(x T ) ( ) T ξ = g X s ds Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

40 Error Analysis Convergence Resuls Error coming from a finie number of chaos We invesigae now E q,p relaed o Y q Y q,p. Proposiion Under h regulariy assumpion E q+1,p C T (T + 1)L 2 f E q,p + K p + 1 where K depends on he derivaives of f and ξ and T. Corollary We have E q,p A p + 1, A = K (C T (T + 1)L 2 f )q 1 C T (T + 1)L 2 f 1. Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

41 Error Analysis Convergence Resuls Error coming from a finie number of chaos We deduce from his resul ha lim q + lim Y q,p = Y. p + When T is small, C T (T + 1)L 2 f < 1, A does no depend on q and lim p + lim Y q,p = Y. q + The algorihm breaks he srucure of BSDEs Le us wrie (Y q+1,p, Z q+1,p ) in a backward way ( T = C p (ξ) + C p f ( s, Ys q,p Y q+1,p T + f (s, Y q,p s, Zs q,p )ds ), Zs q,p ) T ds T Zs q+1,p db s f ( s, Y q,p s, Zs q,p ) ds Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

42 Error Analysis Convergence Resuls Error coming from a finie number of chaos If lim q Y q,p exiss, hen one has Y,p = C p (ξ) + C p ( T T + f (s, Y,p s f ( s, Y,p s T, Zs,p )ds ), Zs,p ) T ds Zs,p db s f ( s, Y,p s The erminal condiion involves all he rajecory of he soluion. We were able o solve his BSDE only when T is small, Zs,p ) ds Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

43 Error Analysis Error coming from he runcaion of he basis Error coming from he runcaion of he basis Proposiion Under he regulariy assumpion, ) 2αξ 1 E q+1,p,n C 1 T (T + 1)L 2 f E q,p,n + K 1 ( T N where C 1 is a scalar and K 1 depends on he derivaives of f and ξ. Moreover, E q,p,n A 1 ( T N ) 2αξ 1, A 1 = K 1 T (T + 1)e T (C 1T (T + 1)L 2 f )q 1 C 1 T (T + 1)L 2 f 1 As before, he consan T (T + 1) appears Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

44 Error Analysis Global error Proposiion Under he regulariy assumpion, wih Convergence resul E C2 q + A ( ) 2αξ 1 T p A 1 N A = K (C T (T + 1)L 2 f )q 1 C T (T + 1)L 2 f 1 A 1 = K 1 T (T + 1)e T (C 1T (T + 1)L 2 f )q 1 C 1 T (T + 1)L 2 f 1 Corollary In paricular, lim lim q p lim Y q,p,n = Y N Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

45 Error Analysis Global error Perspecives Try o weaken he regulariy assumpion Undersand for large p and T he BSDE Y,p = C p (ξ) + C p ( T T + f (s, Y,p s f ( s, Y,p s T, Zs,p )ds ), Zs,p ) T ds Zs,p db s f ( s, Y,p s This mehod works also for he dynamic programing equaion, Zs,p ) ds Y = E (Y + h F ) + hf (, Y, Z ), Z = h 1 E (Y +h (B +h B ) F ) Numerical simulaions are no so good Error analysis Ph. Briand & C. Labar (Univ. Savoie) Simulaion of BSDEs and Wiener Chaos Expansions Rennes, May 22-24, / 46

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Simulation of BSDEs and. Wiener Chaos Expansions

Simulation of BSDEs and. Wiener Chaos Expansions Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp://www.lama.univ-savoie.fr/ Sochasic Analysis Seminar Oxford, June 1, 213 Inroducion