Discrete time approximation of BSDEs with Lipschitz coefficients B. - PowerPoint PPT Presentation

Discrete time approximation of BSDEs with Lipschitz coefficients B. Bouchard Ceremade, University Paris-Dauphine Joint works with: N. Touzi, R. Elie, J.-F. Chassagneux and S. Menozzi

BSDEs and PDEs

Semilinear parabolic PDEs: formal link • The solution ( Y, Z ) of � T � T Y t = g ( X T ) + f ( X s , Y s , Z s ) ds − Z s dW s , t t is related to the solution u of −L u − f ( · , u, Duσ )=0 on [0 , T ) × R d , u ( T, · )= g on R d through “ Z t = Du ( t, X t ) σ ( X t ) ′′ Y t = u ( t, X t ) and ⇒ Two point of views : Solve the pde to approximate ( Y, Z ) / Solve the BSDE to approximate u . Remark: BSDEs can be defined in a non Markovian setting ⇒ non Marko- vian extension of PDEs.

Numerical resolution: first approaches • Ma, Protter and Yong (94), Douglas, Ma and Protter (96), Ma, Protter, San Martin and Torres (02): Du ) and set ( Y π , Z π ) = (ˆ Duσ )( · , X π ). u, ˆ u, ˆ solve the PDE ⇒ (ˆ • Coquet, Mackevicius and Memin (98), Briand, Delyon and Memin (01), Antonelli and Kohatsu (00): approximate W by a discrete random walk (with values in a finite state- space) and solve the associated discrete time BSDE ( ∼ tree method). ⇒ Curse of dimensionality !

Euler scheme approximation

The forward process X • Fix a grid of [0 , T ]: π := { t i := hi, i ≤ n } with h = T/n . • Set X π 0 = X 0 • For i = 1 , . . . , n , set X π X π t i − 1 + b ( X π t i − 1 ) h + σ ( X π = t i − 1 )( W t i − W t i − 1 ) t i

The forward process X • Fix a grid of [0 , T ]: π := { t i := hi, i ≤ n } with h = T/n . • Set X π 0 = X 0 • For i = 1 , . . . , n , set X π X π t i − 1 + b ( X π t i − 1 ) h + σ ( X π = t i − 1 )( W t i − W t i − 1 ) t i • Error: 1   2 1 | X t − X π t i | 2 2 . max sup i<n E ≤ Ch   t ∈ [ t i ,t i +1 ]

The BSDE ( Y, Z ) : Adapted backward Euler scheme • For i = n − 1 , . . . , 0, write Y t i ∼ Y t i +1 + f ( X t i , Y t i , Z t i ) h − Z t i ( W t i +1 − W t i ) (1) � � and take E · | F t i to get � � Y t i ∼ E Y t i +1 | F t i + f ( X t i , Y t i , Z t i ) h

The BSDE ( Y, Z ) : Adapted backward Euler scheme • For i = n − 1 , . . . , 0, write Y t i ∼ Y t i +1 + f ( X t i , Y t i , Z t i ) h − Z t i ( W t i +1 − W t i ) (2) � � and take E · | F t i to get � � Y t i ∼ E Y t i +1 | F t i + f ( X t i , Y t i , Z t i ) h multiply (2) by ( W t i +1 − W t i ) Y t i ( W t i +1 − W t i ) ∼ Y t i +1 ( W t i +1 − W t i ) + f ( X t i , Y t i , Z t i )( W t i +1 − W t i ) h − Z t i ( W t i +1 − W t i )( W t i +1 − W t i ) � � and take E · | F t i � � 0 Y t i +1 ( W t i +1 − W t i ) | F t i ∼ E − Z t i h

The BSDE ( Y, Z ) : Adapted backward Euler scheme (2) • Recall: � � Y t i ∼ E Y t i +1 | F t i + f ( X t i , Y t i , Z t i ) h � � 0 ∼ E Y t i +1 ( W t i +1 − W t i ) | F t i − Z t i h

The BSDE ( Y, Z ) : Adapted backward Euler scheme (2) • Recall: � � Y t i ∼ E Y t i +1 | F t i + f ( X t i , Y t i , Z t i ) h � � 0 ∼ E Y t i +1 ( W t i +1 − W t i ) | F t i − Z t i h • Set Y π T = g ( X π T ) and for i = n − 1 , . . . , 0 � � Y π Y π + f ( X π t i , Y π t i , Z π = E t i +1 | F t i t i ) h t i where � � Z π h − 1 E Y π = t i +1 ( W t i +1 − W t i ) | F t i t i

The BSDE ( Y, Z ) : Adapted backward Euler scheme (2) • Recall: � � Y t i ∼ E Y t i +1 | F t i + f ( X t i , Y t i , Z t i ) h � � 0 ∼ E Y t i +1 ( W t i +1 − W t i ) | F t i − Z t i h • Set Y π T = g ( X π T ) and for i = n − 1 , . . . , 0 � � Y π Y π + f ( X π t i , Y π t i , Z π = E t i +1 | F t i t i ) h t i where � � Z π h − 1 E Y π = t i +1 ( W t i +1 − W t i ) | F t i t i • Could alternatively set � � � � Y π Y π f ( X π t i , Y π t i +1 , Z π = E t i +1 | F t i + E t i ) | F t i h t i

Numerical implementation

Quantization • Bally, Pages and Printems for the case f independent of Z . • Replace X π by a quantized version ˆ X π taking a finite number of possible values. X π . • Estimate the transition probabilities of ˆ Y π X π • Use the algorithm: ˆ T = g ( ˆ T ) and for i = n − 1 , . . . , 0 � � Y π Y π X π X π Y π ˆ ˆ t i +1 | ˆ + f ( ˆ t i , ˆ = E t i ) h t i t i

Pure Monte-Carlo approaches • Simulate ( X π,j , W j , j ≤ N ) Y π,j = g ( X π,j • Set ˆ T ) T • Given ˆ E an approximation of E based on the simulated data, use the induction � � Y π,j t i +1 | X π,j + f ( X π,j Y π,j Z π,j Y π ˆ ˆ ˆ t i , ˆ , ˆ = t i ) h E t i t i t i � � Z π,j t i +1 ( W t i +1 − W t i ) | X π,j h − 1 ˆ Y π ˆ ˆ = E t i t i • Two alternatives : 1. Chevance (97), Longstaff and Schwartz (01), Clement, Lamberton and Protter (02), Gobet, Lemor and Warin (05): non-parametric regression. 2. Lions and Regnier (01), B., Ekeland and Touzi (04), B. and Touzi (04): Malliavin calculus approach to rewrite conditional expectations in terms of unconditional expectations.

Approximation error

Control of the approximation error • Say f ≡ 0, then � T � T Y t i = g ( X T ) + f ( X s , Y s , Z s ) ds − Z s dW s t i t i � t i +1 = Y t i +1 − Z s dW s t i implies � � = E Y t i Y t i +1 | F t i .

Control of the approximation error • Say f ≡ 0, then � T � T Y t i = g ( X T ) + f ( X s , Y s , Z s ) ds − Z s dW s t i t i � t i +1 = Y t i +1 − Z s dW s t i implies � � = E Y t i Y t i +1 | F t i . Thus   � t i | 2 � � | Y t i +1 − Y t i | 2 � t i | 2 | Y t − Y π | Y t i +1 − Y π max sup ≥ max ≥ max i<n E i<n E i<n E   t ∈ [ t i ,t i +1 ]   | Y t − Y t i | 2  =: c R ( Y ) 2 ≥ c max sup i<n E  S 2 t ∈ [ t i ,t i +1 ] for some c > 0.

Control of the approximation error (2) • Set �� t i +1 � Z t i := h − 1 E ˜ Z s ds | F t i t i then     � t i +1 � t i +1 � � � Z t − Z π t i � 2 dt Z t i � 2 dt  =: R ( Z ) 2  ≥ E � Z t − ˜ E H 2 t i t i i i

Control of the approximation error (3) • Conclusion: up to a constant c > 0, the error 1 1     � t i +1 2 2 � t i | 2 t i � 2 dt | Y t − Y π � Z t − Z π Err( h ) := max sup + E i<n E    t i t ∈ [ t i ,t i +1 ] i is bounded from below by 1 1     � t i +1 2 2 � | Y t − Y t i | 2 Z t i � 2 dt � Z t − ˜ R ( Y ) S 2 + R ( Z ) H 2 = max sup + E i<n E    t i t ∈ [ t i ,t i +1 ] i

Control of the approximation error (3) • Conclusion: up to a constant c > 0, the error 1 1     � t i +1 2 2 � t i | 2 t i � 2 dt | Y t − Y π � Z t − Z π Err( h ) := max sup + E i<n E    t i t ∈ [ t i ,t i +1 ] i is bounded from below by 1 1     � t i +1 2 2 � | Y t − Y t i | 2 Z t i � 2 dt � Z t − ˜ R ( Y ) S 2 + R ( Z ) H 2 = max sup + E i<n E    t i t ∈ [ t i ,t i +1 ] i • One can actually show that � � 1 Err( h ) = O R ( Y ) S 2 + R ( Z ) H 2 + h 2

Control of the approximation error (4) • Thus � � 1 Err( h ) = O R ( Y ) S 2 + R ( Z ) H 2 + h 2 where (formally) R ( Y ) 2 | 2 ] S 2 = max i<n E [ sup | u ( t, X t ) − u ( t i , X t i ) � �� � t ∈ [ t i ,t i +1 ] � �� � Y t Y ti and � t i +1 �� t i +1 � � R ( Z ) 2 − h − 1 E � 2 dt ] H 2 = E [ � Duσ ( t, X t ) Duσ ( s, X s ) | F t i t i t i � �� � i Z t � �� � ˜ Z ti • The error depends on a very weak notion of regularity of ( u, Du ).

Regularity results

Semilinear PDEs • Theorem (Ma and Zhang 02, and B. and Touzi 04): Assume all the coefficients are Lipschitz continuous. Then, 1 1 R ( Y ) S 2 + R ( Z ) H 2 = O ( h 2 ) and Err( h ) = O ( h 2 )

Semilinear PDEs • Theorem (Ma and Zhang 02, and B. and Touzi 04): Assume all the coefficients are Lipschitz continuous. Then, 1 1 R ( Y ) S 2 + R ( Z ) H 2 = O ( h 2 ) and Err( h ) = O ( h 2 ) u is 1 • Remark: 2 -H¨ older in t and Lipschitz in x by propagation of the Lipschitz continuity of the terminal condition g . 1 ⇒ Since Y t = u ( t, X t ), R ( Y ) S 2 = O ( h 2 ) corresponds to the fact that X 1 2 -H¨ older in t .

Semilinear PDEs • Theorem (Ma and Zhang 02, and B. and Touzi 04): Assume all the coefficients are Lipschitz continuous. Then, 1 1 R ( Y ) S 2 + R ( Z ) H 2 = O ( h 2 ) and Err( h ) = O ( h 2 ) • Elements of proof for R ( Z ) H 2 : (case f = 0, d = 1, smooth coefficients) Y t = u ( t, X t ) = E [ g ( X T ) | F t ] ∂ u ( t, X t )( ∂ X t ) − 1 σ ( X t ) Z t = Du ( t, X t ) σ ( X t ) = ∂X 0 ∂X 0

Semilinear PDEs • Theorem (Ma and Zhang 02, and B. and Touzi 04): Assume all the coefficients are Lipschitz continuous. Then, 1 1 R ( Y ) S 2 + R ( Z ) H 2 = O ( h 2 ) and Err( h ) = O ( h 2 ) • Elements of proof for R ( Z ) H 2 : (case f = 0, d = 1, smooth coefficients) Y t = u ( t, X t ) = E [ g ( X T ) | F t ] ∂ u ( t, X t )( ∂ X t ) − 1 σ ( X t ) Z t = Du ( t, X t ) σ ( X t ) = ∂X 0 ∂X 0 � � Dg ( X T ) ∂ ( ∂ X t ) − 1 σ ( X t ) = E X T | F t ∂X 0 ∂X 0 � �� � say=1 for simplicity