An approach to fully coupled FBSDEs via functional differential - PowerPoint PPT Presentation

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Existence and uniqueness of solutions Related discretization algorithms for Brownian FBSDEs An approach to fully coupled FBSDEs via functional differential equations Matteo Casserini ∗ joint work with Gechun Liang † ∗ Department of Mathematics ETH Z¨ urich † Oxford-Man Institute Workshop ”New advances in BSDEs for financial engineering applications”, Tamerza October 26, 2010 Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Existence and uniqueness of solutions Related discretization algorithms for Brownian FBSDEs Outline 1 Brownian FBSDEs as functional differential equations 2 Fully coupled forward–backward stochastic dynamics 3 Existence and uniqueness of solutions 4 Related discretization algorithms for Brownian FBSDEs Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 2/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Outline 1 Brownian FBSDEs as functional differential equations 2 Fully coupled forward–backward stochastic dynamics 3 Existence and uniqueness of solutions 4 Related discretization algorithms for Brownian FBSDEs Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 3/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Introduction Aim: Introduce a forward approach for a general class of fully coupled FBSDEs Result: System of forward equations where the coefficients depend also on the terminal values of the solution Conflict between forward and backward components partly avoided Purely probabilistic (random coefficients) Allows to treat other types of non–classical forward–backward equations Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 4/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Motivating observation ( Y t ) 0 ≤ t ≤ T a semimartingale on (Ω , F , ( F t ) 0 ≤ t ≤ T , P ) with known terminal value Y T = ξ ∈ L 1 ( F T ) . Doob-Meyer decomposition: Y t = M t − V t , M martingale, V cont. adapted process of finite variation. If V T is integrable, then: M t = M ( V , ξ ) t = E [ ξ + V T |F t ] ∀ t ∈ [ 0 , T ] , Y t = Y ( V , ξ ) t = E [ ξ + V T |F t ] − V t ∀ t ∈ [ 0 , T ] . (1.1) Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 5/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Motivating observation ( Y t ) 0 ≤ t ≤ T a semimartingale on (Ω , F , ( F t ) 0 ≤ t ≤ T , P ) with known terminal value Y T = ξ ∈ L 1 ( F T ) . Doob-Meyer decomposition: Y t = M t − V t , M martingale, V cont. adapted process of finite variation. If V T is integrable, then: M t = M ( V , ξ ) t = E [ ξ + V T |F t ] ∀ t ∈ [ 0 , T ] , Y t = Y ( V , ξ ) t = E [ ξ + V T |F t ] − V t ∀ t ∈ [ 0 , T ] . (1.1) Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 5/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Motivating observation ( Y t ) 0 ≤ t ≤ T a semimartingale on (Ω , F , ( F t ) 0 ≤ t ≤ T , P ) with known terminal value Y T = ξ ∈ L 1 ( F T ) . Doob-Meyer decomposition: Y t = M t − V t , M martingale, V cont. adapted process of finite variation. If V T is integrable, then: M t = M ( V , ξ ) t = E [ ξ + V T |F t ] ∀ t ∈ [ 0 , T ] , Y t = Y ( V , ξ ) t = E [ ξ + V T |F t ] − V t ∀ t ∈ [ 0 , T ] . (1.1) Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 5/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Formally: alternative formulation of Brownian FBSDEs Probability space (Ω , F , P ) with a m -dim. BM W ( F t ) 0 ≤ t ≤ T corresponding augmented filtration Classical fully coupled FBSDE of the form � dY t = − f ( t , X t , Y t , Z t ) dt + Z t dW t , Y T = Φ( X T ) , dX t = µ ( t , X t , Y t , Z t ) dt + σ ( t , X t , Y t ) dW t , X 0 = x , (1.2) where f : Ω × [ 0 , T ] × R n × R d × R d × m → R d , µ : Ω × [ 0 , T ] × R n × R d × R d × m → R n , σ : Ω × [ 0 , T ] × R n × R d → R n × m , Φ : Ω × R n → R d . Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 6/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Formally: alternative formulation of Brownian FBSDEs Probability space (Ω , F , P ) with a m -dim. BM W ( F t ) 0 ≤ t ≤ T corresponding augmented filtration Classical fully coupled FBSDE of the form � dY t = − f ( t , X t , Y t , Z t ) dt + Z t dW t , Y T = Φ( X T ) , dX t = µ ( t , X t , Y t , Z t ) dt + σ ( t , X t , Y t ) dW t , X 0 = x , (1.2) where f : Ω × [ 0 , T ] × R n × R d × R d × m → R d , µ : Ω × [ 0 , T ] × R n × R d × R d × m → R n , σ : Ω × [ 0 , T ] × R n × R d → R n × m , Φ : Ω × R n → R d . Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 6/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Introduction Existence and uniqueness of solutions Alternative formulation of Brownian FBSDEs Related discretization algorithms for Brownian FBSDEs Formally: alternative formulation of Brownian FBSDEs Define an associated system of functional differential equations: � dV t = f ( t , X t , Y ( V , X ) t , Z ( V , X ) t ) dt , dX t = µ ( t , X t , Y ( V , X ) t , Z ( V , X ) t ) dt + σ ( t , X t , Y ( V , X ) t ) dW t (1.3) with initial conditions V 0 = 0, X 0 = x , where M ( V , X ) t := E [Φ( X T ) + V T |F t ] , Y ( V , X ) t := E [Φ( X T ) + V T |F t ] − V t , Z ( V , X ) t := D t M ( V , X ) T = D t (Φ( X T ) + V T ) ∀ t ∈ [ 0 , T ] . (1.4) Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 7/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Setting Existence and uniqueness of solutions Fully coupled forward–backward stochastic dynamics Related discretization algorithms for Brownian FBSDEs Outline 1 Brownian FBSDEs as functional differential equations 2 Fully coupled forward–backward stochastic dynamics 3 Existence and uniqueness of solutions 4 Related discretization algorithms for Brownian FBSDEs Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 8/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Setting Existence and uniqueness of solutions Fully coupled forward–backward stochastic dynamics Related discretization algorithms for Brownian FBSDEs Setting (Ω , F , P ) probability space with a m -dim. BM W , ( F t ) 0 ≤ t ≤ T with usual assumptions C ([ 0 , T ] , R d ) := { V : Ω × [ 0 , T ] → R d | V continuous and adapted, E [ max j sup t | V j t | 2 ] < ∞} C 0 ([ 0 , T ] , R d ) := C ([ 0 , T ] , R d ) ∩ { V | V 0 = 0 } M 2 ([ 0 , T ] , R d ) := { M : Ω × [ 0 , T ] → R d | M square integrable martingale on [ 0 , T ] } Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 9/32

Brownian FBSDEs as functional differential equations Fully coupled forward–backward stochastic dynamics Setting Existence and uniqueness of solutions Fully coupled forward–backward stochastic dynamics Related discretization algorithms for Brownian FBSDEs Setting � � V � C [ 0 , T ] := E [ sup 0 ≤ t ≤ T | V t | 2 ] S ([ 0 , T ] , R d ) := C ([ 0 , T ] , R d ) ⊕ M 2 ([ 0 , T ] , R d ) H 2 ([ 0 , T ] , R p ) := { Z : Ω × [ 0 , T ] → R p | Z predictable, � T � Z � 2 0 | Z t | 2 dt ] < ∞} H 2 [ 0 , T ] := E [ Matteo Casserini (Gechun Liang) Fully coupled BSDEs: a functional differential approach 10/32