Backward stochastic dynamics on a filtered probability space

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1 Backward sochasic dynamics on a filered probabiliy space Gechun Liang Oxford-Man Insiue, Universiy of Oxford based on join work wih Terry Lyons and Zhongmin Qian Page 1 of 15 gliang@oxford-man.ox.ac.uk

2 Backward sochasic differenial equaion (BSDE) revisied: Pardoux and Peng (199) On a complee filered probabiliy space (Ω, F, {F }, P ): { dy = f(, Y, Z )d + Z dw, Y T = ξ F T. (1) {F } is generaed by a Brownian moion W. Consrain: Y is adaped o {F }. A soluion is a pair (Y, Z). If f(, y, z) is Lipschiz coninuous w.r.. y and z, here exiss a unique square-inegrable soluion pair (Y, Z) S([, T ]; R) H 2 ([, T ]; R). S([, T ]; R): he space of F -adaped processes wih he norm: Y C[,T ] = E sup [,T ] Y 2 H 2 ([, T ]; R): he space of predicable processes wih he norm: Z H 2 [,T ] = E Z s 2 ds Page 2 of 15

3 BSDE revisied: Pardoux and Peng (199) (Y, Z) is a soluion o BSDE (1) if Y = ξ + f(s, Y s, Z s )ds Z s dw s Idea of he proof: For any fixed (Y (1), Z(1)) S([, T ]; R) H 2 ([, T ]; R), by he maringale represenaion, Y = ξ + f(s, Y s (1), Z s (1))ds admis a unique soluion pair (Y (2), Z(2)). Z s dw s (2) Define a mapping L on S([, T ]; R) H 2 ([, T ]; R) by he linear BSDE (2). L is a conracion mapping. Maringale represenaion + conracion mapping. (hey are coupled ogeher.) Page 3 of 15

4 Poenial mehod for nonlinear PDE On an open bounded subse Ω of R d, { u = f(u) in Ω, u = g on Ω. (3) Poenial mehod: For any fixed u in some appropriae space, { v = f(u) in Ω, v = g on Ω. (4) The Green s represenaion: v(x) = Ω f(u(y))g Ω (x, y)dy = G Ω ν(x) + Ω Ω g(y)µ Ω (x, dy) g(y) G Ω n (x, y)ds y where G Ω ν is he poenial of ν wih dν = f(u)dy, and µ Ω (x, ) is he harmonic measure relaive o x wih µ Ω (x, A) = G Ω A n (x, y)ds y. Define a mapping L by he Poisson equaion (4). L is a conracion mapping. Page 4 of 15

5 Poenial mehod for BSDE An analogy beween superharmonic funcion and semimaringale: Riesz decomposiion: superharmonic funcion = poenial + harmonic funcion Doob-Meyer decomposiion: semimaringale = finie variaion process + maringale Lemma 1. If Y is a semimaringale on (Ω, F, {F }, P ), which saisfies he usual condiions, wih a decomposiion: Y = M V [, T ] where M is an F -adaped maringale, and V is a coninuous and F - adaped finie variaion process, hen M = E(Y T + V T F ) Page 5 of 15 and Y = E(Y T + V T F ) V.

6 Poenial mehod for BSDE Given he erminal daa Y T = ξ and he finie variaion par V, here is an one-o-one correspondence: (ξ, V ) Y (ξ, V ); (ξ, V ) M(ξ, V ) If {F } is generaed by a Brownian moion W, by he maringale represenaion, here exiss a densiy process Z H 2 ([, T ]; R) such ha Z(ξ, V ) s dw s = E(ξ + V T F ) E(ξ + V T ) Translae BSDE (1) ino a funcional differenial equaion: V = f(s, Y (ξ, V ) s, Z(ξ, V ) s )ds (5) Y is deermined by Y (ξ, V ) = E(ξ + V T F ) V. Z is deermined by Z(ξ, V ) s dw s = E(ξ + V T F ) E(ξ + V T ). Page 6 of 15

7 Poenial mehod for BSDE By he conracion mapping, he funcional differenial equaion (5) admis a unique soluion V C([, T ]; R). C([, T ]; R): he space of coninuous and F -adaped finie variaion processes wih he norm: V C[,T ] = E sup [,T ] V 2 (Y, Z) saisfies BSDE (1): i.e. Y = E(ξ + V T F ) Y = ξ + E(ξ + V T F ) (ξ + V T ) + = ξ Z s dw s + f(s, Y s, Z s )ds f(s, Y s, Z s )ds f(s, Y s, Z s )ds Conracion mapping and maringale represenaion are decoupled. Brownian filraion and maringale represenaion are no essenial. Page 7 of 15

8 Backward sochasic dynamics On a filered probabiliy space (Ω, F, {F }, P ), which saisfies he usual condiions, { dy = f(, Y, L(M) )d + dm, (6) Y T = ξ F T. M is a square-inegrable maringale, and Y is adaped o {F }. A soluion is a pair (Y, M). A soluion o he backward sochasic dynamics (6) is a pair (Y, M) saisfying Y = ξ + f(s, Y s, L(M) s )ds + M M T Le Y be a semimaringale, and M be a square-inegrable maringale. For any τ [, T ], a pair (Y, M) is called a sric soluion o he backward sochasic dynamics (6) on [τ, T ], if V = M Y C([τ, T ]; R) such ha V τ = and M = E(ξ + V T F ). V is a fixed poin of L on C([τ, T ]; R) where Page 8 of 15 L(V ) = τ f(s, Y (ξ, V ) s, L(M(ξ, V )) s )ds.

9 Admissible operaor L M 2 ([, T ]; R): he space of square-inegrable maringales wih he norm: M C[,T ] = E sup [,T ] M 2 An operaor L : M 2 ([, T ]; R) H 2 ([, T ]; R) (resp. called admissible if L saisfies he resricion propery. C([, T ]; R)) is L : M 2 ([, T ]; R) H 2 ([, T ]; R) (resp. C([, T ]; R)) is bounded and Lipschiz coninuous by a consan C 1. Examples of L: L(M) = M, M, where M, M is he coninuous par of he quadraic variaion process [M, M]. Suppose {F } is generaed by a Brownian moion W. L(M) = Z, where Z is he densiy represenaion of M. Page 9 of 15

10 Local exisence on [τ, T ] Lemma 2. If here is a consan C 2 such ha f(, y, z) C 2 (1 + + y + z ) and f(, y, z) f(, y, z ) C 2 ( y y + z z ), hen L on C([τ, T ]; R) admis a unique fixed poin provided ha ( ) 2 1 T τ = l ( ) 1. 4C C1 Tha is, he funcional differenial equaion V soluion in C([τ, T ]; R). = L(V ) admis a unique Idea of he proof: sandard use of he fixed poin heorem o L. The sric soluion o (6) on [τ, T ] is: M = E(ξ + V T F ) and Y = M V Page 1 of 15

11 Choose he finie pariion: Global exisence on [, T ] Λ : T T > T 1 > > T k such ha he mesh Λ = max 1 j k T j 1 T j l. For [T j, T j 1 ], 1 j k, define Y (V ()) T = ξ, (L j V ) = where T j f (s, Y j (V ) s, L(M j (V )) s )ds M j (V ) = E [ ] Y j 1 (V (j 1)) Tj 1 + V Tj 1 F, Y j (V ) = M j (V ) V Noe ha a he pariion poins T j 1 for 2 j k, Y j 1 (V (j 1)) Tj 1 = Y j (V (j)) Tj 1 Page 11 of 15 V (j 1) Tj 1 V (j) Tj 1

12 Exisence and uniqueness heorem For 1 j k, consruc (Y, M) as Y = Y (j) if [T j, T j 1 ] and define V by shifing i a he pariion poins: V (k) if [, T k 1 ], V (k 1) + V (k) Tk 1 if [T k 1, T k 2 ], V = V (1) + k l=2 V (l) T l 1 if [T 1, T ]. Then, i is easy o see ha V C([, T ]; R). Finally we define M = Y V for [, T ]. Theorem 1. There exiss a unique V C([, T ]; R) such ha V = f(s, Y s, L(M) s )ds where M = E(ξ + V T F ) and Y = M V. Moreover (Y, M) saisfies: Page 12 of 15 Y = ξ + f(s, Y s, L(M) s )ds + M M T

13 Examples of backward sochasic dynamics Suppose {F } is generaed by a Brownian moion W. By he maringale represenaion, here exiss a densiy process Z H 2 ([, T ]; R) such ha M = E(M ) + Z s dw s. Define L(M) = M, M = Z s 2 ds hen he backward sochasic dynamics (6) becomes ( ) dy = f, Y, Z s 2 ds d + Z dw, Y T = ξ. (7) Page 13 of 15 Define L(M) = Z hen he backward sochasic dynamics (6) becomes { dy = f(, Y, Z )d + Z dw, Y T = ξ. (8)

14 Commens Backward sochasic dynamics is a generic exension of a class of nonlinear PDE ino infinie dimensional pah space. some nonlinear PDE is a pahwise version of backward sochasic dynamics. Backward dynamics under oher consrains (no adapeness consrains): (ξ, V ) Y (ξ, V ); (ξ, V ) M(ξ, V ) are general projecions (no condiional expecaions). Exensions o more general case: dy = f (, Y, L(M) )d R\{} Y T = ξ F T. N i=1 f i (, Y )dw i f N+1 (, Y, z)(n(, dz) ν(dz)) + dm, (9) Page 14 of 15

15 Thank you! Page 15 of 15