Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example A second order discretization and efficient simulation for Backward SDEs Konstantinos Manolarakis Imperial College London joint with Dan Crisan New advances in Backward SDEs for financial engineering applications October 2010, Tamerza K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Forward Backward SDEs and related PDEs. Second order discretization. Simulation with the cubature method Numerical examples. K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example ( W t , F t ) 0 ≤ t ≤ T a d -dimensional BM on (Ω , F , P ) . Let ( X , Y , Z ) = { ( X t , Y t , Z t ) 0 ≤ t ≤ T } ∈ R d × R × R d be the solution of the (decoupled) system: � t � t d � V i ( X s ) ◦ dW i X t = x + V 0 ( X s ) ds + s , 0 0 i = 1 (1) � T � T f ( X s , Y s , Z s ) ds − Z s · dW s , Y t = Φ( X T ) + t t - V i : R d → R d smooth vector fields. - Φ( X T ) called the final condition - f : R d × R × R d → R Lipschitz, called ”the driver”. K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Let u ∈ C 1 , 2 ([ 0 , T ) × R d ) be the solution of the final value Cauchy problem L 0 + L � � t ∈ [ 0 , T ) , x ∈ R d � u = − f ( t , x , u , ( ∇ uV ) ( x )) , , (2) x ∈ R d u ( T , x ) = Φ( x ) , where L i � 2 . � d L is the second order differential operator L = 1 � 2 i = 1 L i , i = 0 , 1 ..., d , are the first order differential operators associated to V i = ( V j i ) d j = 1 . d d L i = L 0 = ∂ t + � V j � V j i ∂ x i , 0 ∂ x j j = 1 j = 1 V is the matrix V = ( V 1 , ..., V d ) . K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Theorem (Peng 1991,1992, Pardoux & Peng 1992) The (viscosity or classical ) solution of (2) admits the following Feynman-Kac representation � T � � u ( t , x ) = Y t , x Φ( X t , x f ( s , X t , x s , Y t , x s , Z t , x = E T ) + s ) ds , (3) t t where ( X t , x , Y t , x , Z t , x ) is the ‘stochastic flow’ associated to (1) d dX t , x = V 0 ( X t , x � V i ( X t , x s ) ◦ dW i s ∈ [ t , T ] , s ) ds + s , s i = 1 (4) dY t , x = − f ( X t , x s , Y t , x s , Z t , x s ) ds + Z t , x · dW s . s s X t , x Y t , x = Φ( X t , x = x , T ) t T b ( R d ) then Z t , x = ∇ u ( s , X t , x s ) V ( X t , x If in addition u ∈ C 1 s ) . s K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Notation: We fix a a partition π := { 0 = t 0 < t 1 < . . . < t n } . A is the set of multi indices A := ∪ j { 0 , 1 , . . . , d } j . Norm on multi indices | α | = length of α, � α � := | α | + card { j : α j = 0 , 1 ≤ j ≤ | α |} . For α = ( α 1 , . . . , α k ) ∈ A , we denote by L α u := L α 1 . . . L α k u and the iterated Stratonovich integral for appropriate g : [ 0 , T ] × R d → R : g ( s , X s ) | β | = 0 � s J β [ g ( · , X · )] t , s := t J β − [ g ( · , X · )] t , u du l ≥ 1 , j l = 0 � s t J β − [ g ( · , X · )] t , u ◦ dW j l ( u ) l ≥ 1 , j l � = 0 . K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example We can develop the Stratonovich-Taylor expansion for appropriate g : [ 0 , T ] × R d → R : : g ( t , X 0 , x � L α g ( 0 , x ) J α [ 1 ] 0 , t + � J α [ L α g ( · , X · )] 0 , t ) = t � α �≤ m � α � = m + 1 , m + 2 = Tayl ( g , t ) + R m ( t , g ) (5) R m ( t , g ) is called the remainder. E [ | R m ( t , g ) | ] = O ( t ( m + 1 ) / 2 ) , t < 1 K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example First revisit the Bouchard-Touzi-Zhang discretization (Euler style). Assume that we “know” { X } n i = 0 : Y π, 1 Z π, 1 := Φ( X n ) , = 0 t n t n 1 � � Z π, 1 Y π, 1 := E i t i + 1 ∆ W i + 1 t i δ i + 1 � � � � Y π, 1 Y π, 1 X i , Y π, 1 , Z π, 1 := E i + δ i + 1 f , i = n − 1 , . . . , 0 . t i t i + 1 t i t i where δ i + 1 = t i + 1 − t i , ∆ W i + 1 = W t i + 1 − W t i . K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Theorem (Bouchard and Touzi(2004), Zhang(2004), Gobet and Labart(2007)) When coefficients of the FBSDE are smooth � � 2 2 � � � � � Y t i − Y π, 1 � Z t i − Z π, 1 = O ( � π � 2 ) 0 ≤ i ≤ n − 1 E max + δ i + 1 � � � � t i t i � � whereas, if these are Lipschitz continuous � � 2 2 � � � � � Y t i − Y π, 1 � Z t i − Z π, 1 max + δ i + 1 = O ( � π � ) 0 ≤ i ≤ n − 1 E � � � � t i t i � � K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Towards the second order scheme: For the driver, we use the Trapezoid rule rather than Euler. Assume that we also “know” Z i = ∇ u ( t i , X t i ) V ( X t i ) : + δ i + 1 � � � � � � Y π, 2 Y π, 2 f ( X t i , Y π, 2 f ( X t i + 1 , Y π, 2 := E i , Z t i ) + E i t i + 1 , Z t i + 1 ) t i t i + 1 t i 2 With standard arguments using the Stratonovich Taylor expansions, we can show : � � � Y t i − Y π, 2 � � t i � � t i + 1 f (Θ s ) ds − δ i + 1 � � ��� � � � Y t i + 1 − Y π, 2 f (Θ π, 2 f (Θ π, 2 = t i + 1 + ) + E i t i + 1 ) � E i � � t i 2 � t i � � � � Y t i + 1 − Y π, 2 � + δ 3 � α �≤ 4 � L α u ( t i + 1 , · ) � ∞ ≤ ( 1 + C δ i + 1 ) � E i i + 1 max � � t i + 1 where Θ t = ( X t , Y t , Z t ) , Θ π, 2 := ( X t i , Y π, 2 , Z π, 2 ) . t i t i t i K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Higher order for Z : We fix l = 1 , . . . , d and look at the BTZ approximation using the Taylor expansion: K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Higher order for Z : We fix l = 1 , . . . , d and look at the BTZ approximation using the Taylor expansion: 1 1 � � � � Y t i + 1 ∆ W l u ( t i + 1 , X t i + 1 )∆ W l E i = E i i + 1 i + 1 δ i + 1 δ i + 1 1 1 � � ∆ W l � L α u ( t i , X t i ) J α [ 1 ] t i , t i + 1 + R 4 ( u , δ i + 1 )∆ W l = E i E i i + 1 i + 1 δ i + 1 δ i + 1 � α �≤ 4 K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Higher order for Z : We fix l = 1 , . . . , d and look at the BTZ approximation using the Taylor expansion: 1 1 � � � � Y t i + 1 ∆ W l u ( t i + 1 , X t i + 1 )∆ W l E i = E i i + 1 i + 1 δ i + 1 δ i + 1 1 1 � � ∆ W l � L α u ( t i , X t i ) J α [ 1 ] t i , t i + 1 + R 4 ( u , δ i + 1 )∆ W l = E i E i i + 1 i + 1 δ i + 1 δ i + 1 � α �≤ 4 t i + 1 � � = Z l � L α u ( t i , X t i ) E i ∆ W l i + 1 J α [ 1 ] t i , t i + 1 + δ 2 � α � = 5 , 6 � L α u ( t i + 1 , · ) � ∞ i + 1 max δ i + 1 � α � = 3 (6) K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Higher order for Z : We fix l = 1 , . . . , d and look at the BTZ approximation using the Taylor expansion: 1 1 � � � � Y t i + 1 ∆ W l u ( t i + 1 , X t i + 1 )∆ W l E i = E i i + 1 i + 1 δ i + 1 δ i + 1 1 1 � � ∆ W l � L α u ( t i , X t i ) J α [ 1 ] t i , t i + 1 + R 4 ( u , δ i + 1 )∆ W l = E i E i i + 1 i + 1 δ i + 1 δ i + 1 � α �≤ 4 t i + 1 � � = Z l � L α u ( t i , X t i ) E i ∆ W l i + 1 J α [ 1 ] t i , t i + 1 + δ 2 � α � = 5 , 6 � L α u ( t i + 1 , · ) � ∞ i + 1 max δ i + 1 � α � = 3 (6) Since k � J α [ 1 ] t i , t i + 1 J ( l ) [ 1 ] t i , t i + 1 = J ( α 1 ,...,α j − 1 , l ,α j ,...,α k ) [ 1 ] t i , t i + 1 j = 1 J α [ 1 ] t i , t i + 1 � � E i = 0 , if � α � = 1 + 2 N + K Manolarakis Second order algorithm for BSDEs
Synopsis Forward-Backward SDEs Discretization Simulation A numerical example Cubature +recombination Another numerical example Which multi indices α , with � α � = 3, satisfy � � ∆ W l i + 1 J α [ 1 ] t i , t i + 1 E i � = 0 K Manolarakis Second order algorithm for BSDEs
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