Convergence of Bender/Denk algorithm L 2 -error estimates and rate of - PowerPoint PPT Presentation

Assumptions and framework Approximation Numerical examples Convergence of Bender/Denk algorithm L 2 -error estimates and rate of convergence for Monte Carlo projection Plamen Turkedjiev Tuesday 26th October, 2010 Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Approximation Numerical examples Approximate the following BSDE using a discretized scheme: − d Y s = f ( s, X s , Y s , Z s )d s − Z s d W s , 0 ≤ t < T Y T (= ξ ) = Φ( X T ) ◮ ( X t ) 0 ≤ t ≤ T diffusion with Lipschitz data ( b, σ ) , t �→ b ( t, 0) , σ ( t, 0) bounded ◮ ( x, y, z ) �→ f ( t, x, y, z ) and x �→ Φ( x ) are Lipschitz continuous, t �→ f ( t, 0 , 0 , 0) bounded Discretization: Time-grid with N points, ∆ k := t k +1 − t k , | π | := max 0 ≤ k ≤ N − 1 ∆ k , ∆ W t k := W k +1 − W k Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate 1. Select regression basis p ( x ) = ( φ 1 ( x ) , . . . , φ M ( x )) such that t k ) ∈ L 2 and for Euler simulation ( X ( π ) t k ) 0 ≤ k ≤ N , φ i ( X ( π ) p ( X ( π ) t k )[ p ( X ( π ) � t k )] tr � invertible E ∆ W tk p ( X ( π ) √ ∆ k p ( X ( π ) ∈ R M × R M � � Transform v t k := t k ) , t k ) 2. Perform n Picard iterations for regression coefficients Z,t k ) ∈ R M × R M : θ ( n ) ˆ t k = (ˆ θ ( n ) Y,t k , ˆ θ ( n ) 1 Y ( m ) ˆ := ˆ θ ( m ) Y,t j · p ( X ( π ) t j ) , ˆ Z ( m ) θ ( m ) ˆ Z,t j · p ( X ( π ) := √ ∆ k t j ) t j t j N − 1 �� � � ξ ( π ) − θ ( m +1) f ( t j , X ( π ) Y ( m ) Z ( m ) � 2 ˆ � t j , ˆ , ˆ � := arg min E )∆ j − θ · v t k t k t j t j θ j = k 4. Projection estimators: 1 Y ( n ) θ ( n ) Y,t j · p ( X ( π ) Z ( n ) θ ( n ) Z,t j · p ( X ( π ) ˆ := ˆ t j ) , ˆ ˆ := √ ∆ k t j ) t j t j Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate 1. Select regression basis p ( x ) = ( φ 1 ( x ) , . . . , φ M ( x )) such that for Euler simulation ( X ( π ) t k ) 0 ≤ k ≤ N , φ i ( X ( π ) t k ) ∈ L 2 and p ( X ( π ) t k )[ p ( X ( π ) t k )] tr � � invertible E 2. Generate L independent simulations { ( W ( π,l ) ) 0 ≤ k ≤ N } 1 ≤ l ≤ L t k of ( W t k ) 0 ≤ k ≤ N and Euler simulations { ( X ( π,l ) ) 0 ≤ k ≤ N } 1 ≤ l ≤ L t k ∆ W l p ( X ( π,l ) √ ∆ k p ( X ( π,l ) ∈ R M × R M Transform v l � tk � t k := ) , ) t k t k 3. Perform n Picard iterations for regression coefficients θ ( n,L ) θ ( n,L ) θ ( n,L ) Z,t k ) ∈ R M × R M : ˆ = (ˆ Y,t k , ˆ t k   θ ( m,L ) ˆ � � Y ( m,L,l ) θ ( m,L ) ˆ · p ( X ( π,l ) Z ( m,L,l ) Z,t k · p ( X ( π,l ) ˆ , ˆ := ˆ ρ j ) := ˆ ρ j √ ∆ k ) t j t j t j t j Y,t j   L 1 �� θ ( m +1 ,L ) ˆ � � ξ ( π,l ) − := arg min t k L θ l =1 N − 1 � 2 � � f ( t j , X ( π,l ) , ˆ Y ( m,L,l ) , ˆ Z ( m,L,l ) )∆ j − θ · v l � t j t j t j t k j = k Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate 4. Monte Carlo estimators:   θ ( n,L ) ˆ � � Y ( n,L ) θ ( n,L ) Y,t k · p ( X ( π ) Z ( n,L ) Z,t k · p ( X ( π ) ˆ ˆ , ˆ := ˆ ρ k t k ) := ˆ ρ k √ ∆ k t k ) t k t k   Construction of truncation functions: 1. Take R -valued functions ρ ( · ) := max(1 , C 0 | p ( · ) | ) , with C 0 appropriately chosen 2. Choose φ ∈ C 2 b ( R ) such that φ ( x ) 1 [ − 3 / 2 , 3 / 2] ( x ) = x , || φ || ∞ ≤ 2 , and || φ ′ || ∞ ≤ 1 � � � � X ( π ) x 3. ˆ ρ k ( x ) := ρ · φ t k ρ ( X ( π ) tk ) Properties: ρ k ( √ ∆ k ˆ t k ) = √ ∆ k ˆ Y ( n ) Y ( n ) Z ( n ) Z ( n ) ρ k ( ˆ t k ) = ˆ ◮ ˆ and ˆ t k t k ρ k ( x ) | ≤ 2 ρ ( X ( π ) t k ) ∈ L 2 ( X ( π ) ◮ | ˆ t k ) ∀ x ∈ R , 0 ≤ k ≤ N − 1 ◮ Lipschitz continuous with Lipschitz constant 1 Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate � L Normal equations: Set V L t k := 1 l =1 v l t k [ v l t k ] tr L N − 1 � � �� θ ( n +1) ξ ( π ) − f ( t j , X ( π ) Y ( n ) Z ( n ) v t k [ v t k ] tr � ˆ � t j , ˆ t j , ˆ � = E v t k t j )∆ j E t k j = k L N − 1 � � = 1 ξ ( π,l ) − t k ˆ θ ( n +1 ,L ) � � f ( t j , X ( π,l ) , ˆ Y ( n,L,l ) , ˆ Z ( n,L,l ) V L v l )∆ j t k t k t j t j t j L l =1 j = k Want invertible V L t k : � | V L � v t k [ v t k ] tr � � t k − E | ≤ | π | , A L t 0 := ∀ q = 0 , . . . , D, and 0 ≤ k ≤ N − 1 Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate Task: N − 1 Find estimate for | 2 + Y ( n ) Y ( n,L ) Z ( n ) Z ( n,L ) 0 ≤ k ≤ N − 1 E | ˆ − ˆ � E | ˆ t k − ˆ | 2 ∆ k Err ( n, π, L ) = max t k t k t k k =0 Use estimate to prove convergence and find rate of convergence w.r.t. L Theorem Assume: ◮ ( X ( π ) p ( X ( π ) � � t k ) 0 ≤ k ≤ N − 1 , t k ) 0 ≤ k ≤ N − 1 have fourth moments ◮ | π | < min(1 , ¯ V ) for ¯ v t k [ v t k ] tr � � V := min 0 ≤ k ≤ N − 1 λ MIN ( E ) N − 1 � | ρ ( P ( π ) � � t k ) | 2 1 [ A L Err ( n, π, L ) ≤ C E t 0 ] C k =0 n − 1 � CN CN � � � ( C | π | ) m 1 + e 2 KN � + (1 − | π | ) 2 L + ( ¯ V − | π | ) L m =1 Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate First estimate of the error: N − 1 � � | ρ ( X ( π ) � t k ) | 2 1 [ A L Err ( n, π, L ) ≤ C E + t 0 ] C k =0 � t k ) | 2 � θ ( n,L ) θ ( n ) t 0 (ˆ − ˆ CN 0 ≤ k ≤ N − 1 E max | 1 A L t k Decomposition: θ ( n,L ) θ ( n ) t k ) = B (1 ,n ) + B (2 ,n ) + B (3 ,n ) t 0 (ˆ − ˆ 1 A L t k k k k Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate θ ( n − 1) ˆ Y ( n − 1 ,l ) θ ( n − 1) · p ( X ( π,l ) Z ( n − 1 ,l ) ∆ j · p ( X ( π,l ) Set ˆ := ˆ ) , ˆ Z,tj √ := ) t j t j t j t j Y,t j � t k ] − 1 − E v t k [ v t k ] tr � − 1 � B (1 ,n ) t 0 V L � := [ 1 A L × k N − 1 � � v t k ( ξ ( π ) − t j , X ( π ) Y ( n − 1) Z ( n − 1) � t j , ˆ , ˆ � � E f ∆ j ) t j t j j = k B (2 ,n ) t 0 V L t k ] − 1 × :=[ 1 A L k L N − 1 � 1 � ξ ( π,l ) − � t j , X ( π,l ) Y ( n − 1 ,l ) Z ( n − 1 ,l ) � � , ˆ , ˆ v l � � f ∆ j t k t j t j t j L l =1 j = k N − 1 ��� � � ξ ( π ) − t j , X ( π ) Y ( n − 1) Z ( n − 1) � t j , ˆ , ˆ � � − E v t k f ∆ j t j t j j = k Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate t 0 V L t k ] − 1 [ 1 A L L � N − 1 � B (3 ,n ) t j , X ( π,l ) Y ( n − 1 ,l ) Z ( n − 1 ,l ) � v l � , ˆ , ˆ � � := f t k k t j t j t j L l =1 j = k �� � t j , X ( π,l ) Y ( n − 1 ,L,l ) Z ( n − 1 ,L,l ) , ˆ , ˆ � − f ∆ j t j t j t j Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate k -uniform L 2 -bounds for B ( q,n ) k � � Define � θ � 2 = max 0 ≤ k ≤ N − 1 E | θ k | 2 1 A L t 0 Assume: ◮ ( X ( π ) p ( X ( π ) � � t k ) 0 ≤ k ≤ N − 1 , t k ) 0 ≤ k ≤ N − 1 have fourth moments ◮ | π | < min(1 , ¯ V ) for ¯ � v t k [ v t k ] tr � V := min 0 ≤ k ≤ N − 1 λ MIN ( E ) (a) � ( B (1 ,n ) ) � 2 ≤ C (1 − | π | ) − 2 L − 1 k ) � 2 ≤ C ( λ MIN ( E (b) � ( B (2 ,n ) v t k [ v t k ] tr � ) − | π | ) − 1 L − 1 � k (c) E � ( B (3 ,n ) θ ( n − 1) − ˆ 0 ≤ C | π | ( ¯ V − | π | ) − 1 � (ˆ ) � 2 θ ( n − 1 ,L ) ) 1 A L t 0 � 2 0 , k e − 2 K ( N − k ) | θ k | � � where � θ � 0 := max 0 ≤ k ≤ N − 1 Plamen Turkedijev Humboldt Universit¨ at zu Berlin

Assumptions and framework Projection algorithm Approximation Monte Carlo algorithm Numerical examples Error estimate Proof (c). V − | π | ) � ( B (3 ,n ) ( ¯ ) � 2 0 ≤ k ≤ N − 1 e − 2 K ( N − k ) × 0 ≤ max k   L N − 1 2 � � 1 ( f (1 ,n − 1 ,l ) − f (2 ,n − 1 ,l ) � � � � )∆ j E  1 A L � � j j  L t 0 � � l =1 j = k 0 ≤ k ≤ N − 1 e − 2 K ( N − k ) × ≤ C | π | max N − 1 θ ( n − 1) − ˆ � 2 � � e − 2 K ( N − j ) � (ˆ θ ( n − 1 ,L ) ) 1 A L �� � E t 0 � 0 � j = k θ ( n − 1) − ˆ ≤ C | π |� (ˆ θ ( n − 1 ,L ) ) 1 A L t 0 � 2 0 Plamen Turkedijev Humboldt Universit¨ at zu Berlin