Backward stochastic dynamics on a filtered probability space Home - PowerPoint PPT Presentation

Backward stochastic dynamics on a filtered probability space Home Page Title Page ◭◭ ◮◮ Gechun Liang ◭ ◮ Oxford-Man Institute, University of Oxford Page 1 of 15 based on joint work with Go Back Terry Lyons and Zhongmin Qian Full Screen Close gliang@oxford-man.ox.ac.uk Quit

Backward stochastic differential equation (BSDE) revisited: Pardoux and Peng (1990) • On a complete filtered probability space (Ω , F , {F t } t ≥ 0 , P ) : � dY t = − f ( t, Y t , Z t ) dt + Z t dW t , (1) Y T = ξ ∈ F T . – {F t } t ≥ 0 is generated by a Brownian motion W . – Constraint: Y is adapted to {F t } t ≥ 0 . Home Page – A solution is a pair ( Y, Z ) . Title Page • If f ( t, y, z ) is Lipschitz continuous w.r.t. y and z , there exists a unique square-integrable solution pair ( Y, Z ) ∈ S ([0 , T ]; R ) × H 2 ([0 , T ]; R ) . ◭◭ ◮◮ ◭ ◮ – S ([0 , T ]; R ) : the space of F t -adapted processes with the norm: Page 2 of 15 � | Y t | 2 || Y || C [0 ,T ] = E sup Go Back t ∈ [0 ,T ] Full Screen – H 2 ([0 , T ]; R ) : the space of predictable processes with the norm: Close Quit � � T | Z s | 2 ds || Z || H 2 [0 ,T ] = E 0

BSDE revisited: Pardoux and Peng (1990) • ( Y, Z ) is a solution to BSDE (1) if � T � T Y t = ξ + f ( s, Y s , Z s ) ds − Z s dW s t t • Idea of the proof: Home Page – For any fixed ( Y (1) , Z (1)) ∈ S ([0 , T ]; R ) × H 2 ([0 , T ]; R ) , by the mar- Title Page tingale representation, ◭◭ ◮◮ � T � T Y t = ξ + f ( s, Y s (1) , Z s (1)) ds − Z s dW s (2) ◭ ◮ t t Page 3 of 15 admits a unique solution pair ( Y (2) , Z (2)) . Go Back – Define a mapping L on S ([0 , T ]; R ) × H 2 ([0 , T ]; R ) by the linear BSDE Full Screen (2). L is a contraction mapping. Close – Martingale representation + contraction mapping. (they are coupled to- gether.) Quit

Potential method for nonlinear PDE • On an open bounded subset Ω of R d , � − ∆ u = f ( u ) in Ω , (3) u = g on ∂ Ω . • Potential method: – For any fixed u in some appropriate space, � − ∆ v = f ( u ) in Ω , Home Page (4) v = g on ∂ Ω . Title Page ◭◭ ◮◮ – The Green’s representation: ◭ ◮ � � g ( y ) ∂G Ω v ( x ) = f ( u ( y )) G Ω ( x, y ) dy − n ( x, y ) dS y Page 4 of 15 ∂� Ω ∂ Ω � Go Back = G Ω ν ( x ) + g ( y ) µ Ω ( x, dy ) Full Screen ∂ Ω Close where G Ω ν is the potential of ν with dν = f ( u ) dy , and µ Ω ( x, · ) is the harmonic measure relative to x with µ Ω ( x, A ) = − � ∂G Ω n ( x, y ) dS y . Quit ∂� A – Define a mapping L by the Poisson equation (4). L is a contraction mapping.

Potential method for BSDE • An analogy between superharmonic function and semimartingale: – Riesz decomposition: superharmonic function = potential + harmonic function – Doob-Meyer decomposition: semimartingale = finite variation process + martingale Home Page • Lemma 1. If Y is a semimartingale on (Ω , F , {F t } t ≥ 0 , P ) , which satisfies Title Page the usual conditions, with a decomposition: ◭◭ ◮◮ Y t = M t − V t t ∈ [0 , T ] ◭ ◮ Page 5 of 15 where M is an F t -adapted martingale, and V is a continuous and F t - Go Back adapted finite variation process, then Full Screen M t = E ( Y T + V T |F t ) Close and Quit Y t = E ( Y T + V T |F t ) − V t .

Potential method for BSDE • Given the terminal data Y T = ξ and the finite variation part V , there is an one-to-one correspondence: ( ξ, V ) → Y ( ξ, V ); ( ξ, V ) → M ( ξ, V ) • If {F t } t ≥ 0 is generated by a Brownian motion W , by the martingale repre- sentation, there exists a density process Z ∈ H 2 ([0 , T ]; R ) such that � t Home Page Z ( ξ, V ) s dW s = E ( ξ + V T |F t ) − E ( ξ + V T ) Title Page 0 ◭◭ ◮◮ • Translate BSDE (1) into a functional differential equation: ◭ ◮ � t Page 6 of 15 V t = f ( s, Y ( ξ, V ) s , Z ( ξ, V ) s ) ds (5) 0 Go Back – Y is determined by Y ( ξ, V ) t = E ( ξ + V T |F t ) − V t . Full Screen – Z is determined by Close � t Quit Z ( ξ, V ) s dW s = E ( ξ + V T |F t ) − E ( ξ + V T ) . 0

Potential method for BSDE • By the contraction mapping, the functional differential equation (5) admits a unique solution V ∈ C ([0 , T ]; R ) . – C ([0 , T ]; R ) : the space of continuous and F t -adapted finite variation pro- cesses with the norm: � | V t | 2 || V || C [0 ,T ] = E sup t ∈ [0 ,T ] Home Page • ( Y, Z ) satisfies BSDE (1): Title Page � t Y t = E ( ξ + V T |F t ) − f ( s, Y s , Z s ) ds ◭◭ ◮◮ 0 ◭ ◮ i.e. Page 7 of 15 � T Go Back Y t = ξ + E ( ξ + V T |F t ) − ( ξ + V T ) + f ( s, Y s , Z s ) ds t Full Screen � T � T = ξ − Z s dW s + f ( s, Y s , Z s ) ds Close t t Quit • Contraction mapping and martingale representation are decoupled. Brown- ian filtration and martingale representation are not essential.

Backward stochastic dynamics • On a filtered probability space (Ω , F , {F t } t ≥ 0 , P ) , which satisfies the usual conditions, � dY t = − f ( t, Y t , L ( M ) t ) dt + dM t , (6) Y T = ξ ∈ F T . – M is a square-integrable martingale, and Y is adapted to {F t } t ≥ 0 . – A solution is a pair ( Y, M ) . • A solution to the backward stochastic dynamics (6) is a pair ( Y, M ) satisfy- Home Page ing Title Page � T Y t = ξ + f ( s, Y s , L ( M ) s ) ds + M t − M T ◭◭ ◮◮ t ◭ ◮ • Let Y be a semimartingale, and M be a square-integrable martingale. For Page 8 of 15 any τ ∈ [0 , T ] , a pair ( Y, M ) is called a strict solution to the backward stochastic dynamics (6) on [ τ, T ] , if V = M − Y ∈ C ([ τ, T ]; R ) such that Go Back – V τ = 0 and M t = E ( ξ + V T |F t ) . Full Screen – V is a fixed point of L on C ([ τ, T ]; R ) where Close � t Quit L ( V ) t = f ( s, Y ( ξ, V ) s , L ( M ( ξ, V )) s ) ds. τ

Admissible operator L • M 2 ([0 , T ]; R ) : the space of square-integrable martingales with the norm: � | M t | 2 || M || C [0 ,T ] = E sup t ∈ [0 ,T ] • An operator L : M 2 ([0 , T ]; R ) → H 2 ([0 , T ]; R ) (resp. C ([0 , T ]; R ) ) is Home Page called admissible if Title Page – L satisfies the restriction property. ◭◭ ◮◮ – L : M 2 ([0 , T ]; R ) → H 2 ([0 , T ]; R ) (resp. C ([0 , T ]; R ) ) is bounded and ◭ ◮ Lipschitz continuous by a constant C 1 . Page 9 of 15 • Examples of L : Go Back � – L ( M ) t = � M, M � t , where � M, M � is the continuous part of the quadratic variation process [ M, M ] . Full Screen – Suppose {F t } t ≥ 0 is generated by a Brownian motion W . L ( M ) t = Z t , Close where Z is the density representation of M . Quit

Local existence on [ τ, T ] • Lemma 2. If there is a constant C 2 such that | f ( t, y, z ) | ≤ C 2 (1 + t + | y | + | z | ) and | f ( t, y, z ) − f ( t, y ′ , z ′ ) | ≤ C 2 ( | y − y ′ | + | z − z ′ | ) , then L on C ([ τ, T ]; R ) admits a unique fixed point provided that Home Page � 2 Title Page � 1 T − τ = l ≤ √ ∧ 1 . ◭◭ ◮◮ � � 4 C 2 3 + 3 3 + 2 C 1 ◭ ◮ That is, the functional differential equation V = L ( V ) admits a unique Page 10 of 15 solution in C ([ τ, T ]; R ) . Go Back • Idea of the proof: standard use of the fixed point theorem to L . Full Screen • The strict solution to (6) on [ τ, T ] is: Close M t = E ( ξ + V T |F t ) and Y t = M t − V t Quit

Global existence on [0 , T ] • Choose the finite partition: Λ : T ≡ T 0 > T 1 > · · · > T k ≡ 0 such that the mesh | Λ | = max 1 ≤ j ≤ k | T j − 1 − T j | ≤ l . • For t ∈ [ T j , T j − 1 ] , 1 ≤ j ≤ k , define Y 0 ( V (0)) T 0 = ξ , Home Page � t ( L j V ) t = f 0 ( s, Y j ( V ) s , L ( M j ( V )) s ) ds Title Page T j ◭◭ ◮◮ where ◭ ◮ M j ( V ) t = E � � Y j − 1 ( V ( j − 1)) T j − 1 + V T j − 1 |F t , Page 11 of 15 Y j ( V ) t = M j ( V ) t − V t Go Back Full Screen • Note that at the partition points T j − 1 for 2 ≤ j ≤ k , Close – Y j − 1 ( V ( j − 1)) T j − 1 = Y j ( V ( j )) T j − 1 Quit – V ( j − 1) T j − 1 � = V ( j ) T j − 1

Existence and uniqueness theorem • For 1 ≤ j ≤ k , construct ( Y, M ) as Y t = Y ( j ) t if t ∈ [ T j , T j − 1 ] and define V by shifting it at the partition points:  V ( k ) t if t ∈ [0 , T k − 1 ] ,   V ( k − 1) t + V ( k ) T k − 1 if t ∈ [ T k − 1 , T k − 2 ] ,  V t = · · ·  V (1) t + � k Home Page  l =2 V ( l ) T l − 1 if t ∈ [ T 1 , T ] .  Title Page Then, it is easy to see that V ∈ C ([0 , T ]; R ) . Finally we define ◭◭ ◮◮ M t = Y t − V t for t ∈ [0 , T ] . ◭ ◮ Page 12 of 15 • Theorem 1. There exists a unique V ∈ C ([0 , T ]; R ) such that Go Back � t Full Screen V t = f ( s, Y s , L ( M ) s ) ds 0 Close where M t = E ( ξ + V T |F t ) and Y t = M t − V t . Moreover ( Y, M ) satisfies: Quit � T Y t = ξ + f ( s, Y s , L ( M ) s ) ds + M t − M T t