A Discrete-time approximation for reflected BSDEs related to - PowerPoint PPT Presentation

Introduction A Discretization scheme for RBSDEs Convergence for the obliquely RBSDE A Discrete-time approximation for reflected BSDEs related to “switching problem” J-F Chassagneux ∗ (Universit´ e d’ Evry Val d’Essonne) joint work with R. Elie (P9) et I. Kharroubi (P9) New advances in BSDEs for financial engineering applications - Tamerza, October 28, 2010 (*) The research of the author benefited from the support of the ‘Chaire Risque de cr´ edit’, F´ ed´ eration Bancaire Fran¸ caise. J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction A Discretization scheme for RBSDEs Convergence for the obliquely RBSDE Introduction Obliquely reflected BSDEs Example: Starting and stopping problem Representation using “switched” BSDEs A Discretization scheme for RBSDEs Approximation of the forward SDE Approximation of the RBSDE Stability issue Convergence for the obliquely RBSDE Discretizing the reflection Errors analysis Convergence results J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Reflected BSDEs ◮ For ( b , σ ) : R d → R d × M d Lipschitz ( σ may be degenerate) : � t � t X t = X 0 + b ( X u ) d u + σ ( X u ) d W u 0 0 ◮ ‘Simply’ reflected BSDEs on a boundary l ( X ): � T � T � T ( Z t ) ′ d W t + Y t = g ( X T ) + f ( X t , Y t , Z t ) d t − d K t t t t ( C1 ) Y t ≥ l ( X t ) (constrained value process) � T � � ( C2 ) Y t − l ( X t ) d K t = 0 (“optimality” of K) 0 ◮ Extension: doubly reflected BSDEs, reflected BSDEs in convex domain ֒ → normal reflection J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Geometric framework ◮ Multidimensional value process constrained in a domain C ( d ≥ 2) C = { y ∈ R d | y i ≥ P i ( y ) := max j ( y j − c ij ) } with c ii = 0, inf i � = j c ij > 0, c ij + c jk > c ik → P (oblique projection) is L -lipschitz with L > 1 (euclidean norm) ֒ ◮ example d = 2, oblique direction of reflection J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Obliquely reflected BSDEs ◮ System of reflected BSDEs: for 1 ≤ i ≤ d , � T � T Y i t = g i ( X T ) + f i ( X u , Y i u , Z i ( Z i u ) ′ d W u + K i T − K i u ) d u − t t t ( C1 ) Y t ∈ C (constrained by K ) � � � T Y i t − P i ( Y t ) d K i ( C2 ) t = 0 (’optimality’ of K ) 0 ◮ Hu and Tang 07, Hamadene and Zhang 08 J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Starting and Stopping problem (1) Hamadene and Jeanblanc (01): ◮ Consider e.g. a power station producing electricity whose price is given by a diffusion process X : d X t = b ( X t ) d t + σ ( X t ) d W t ◮ Two modes for the power station: mode 1 : operating, profit is then f 1 ( X t ) d t mode 2 : closed, profit is then f 2 ( X t ) d t → switching from one mode to another has a cost: c > 0 ֒ ◮ Management decide to produce electricity only when it is profitable enough. ◮ The management strategy is ( θ j , α j ) : θ j is a sequence of stopping times representing switching times from mode α j − 1 to α j . ( a t ) 0 ≤ t ≤ T is the state process (the management strategy). J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Starting and Stopping problem (2) ◮ Following a strategy a from t up to T , gives � T � f a s ( X s ) d s − J ( a , t ) = c 1 { t ≤ θ j ≤ T } t j ≥ 0 ◮ The optimization problem is then (at t = 0, for α 0 = 1) Y 1 0 := sup a E [ J ( a , 0)] At any date t ∈ [0 , T ] in state i ∈ { 1 , 2 } , the value function is Y i t . J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Solution ◮ Y is solution of a coupled optimal stopping problem �� τ � Y 1 f (1 , X s ) d s + ( Y 2 t = ess sup τ − c ) 1 { τ< T } | F t E t ≤ τ ≤ T t �� τ � Y 2 f (2 , X s ) d s + ( Y 1 t = ess sup E τ − c ) 1 { τ< T } | F t t ≤ τ ≤ T t ◮ The optimal strategy ( θ ∗ j , α ∗ j ) is given by α ∗ θ ∗ j +1 := inf { s ≥ θ ∗ i ∈{ 1 , 2 } Y i j j | Y = max s − c } s α ∗ j +1 := 1 if α ∗ j = 2 , or 2 if α ∗ j = 1 . J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs System of reflected BSDEs Y is the solution of the following system of reflected BSDEs: � T � T � T Y i ( Z i s ) ′ d W s + d K i t = f ( i , X s ) d s − s , i ∈ { 1 , 2 } , t t t with (the coupling...) Y 1 t ≥ Y 2 t − c and Y 2 t ≥ Y 1 t − c , ∀ t ∈ [0 , T ] and (‘optimality’ of K ) � T � T � � � � Y 1 s − ( Y 2 d K 1 Y 2 s − ( Y 1 d K 2 s − c ) s = 0 and s − c ) s = 0 0 0 J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs Remark: related obstacle problem ◮ On R × [0 , T ) � � − ∂ t u 1 − L u 1 − f 1 , u 1 − u 2 + c min = 0 � � − ∂ t u 2 − L u 2 − f 2 , u 2 − u 1 + c min = 0 u 1 ≥ u 2 − c and u 2 ≥ u 1 − c ◮ Terminal condition u ( T , . ) = 0 ◮ Link via Y 1 t = u 1 ( t , X t ) and Y 2 t = u 2 ( t , X t ) J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Obliquely reflected BSDEs A Discretization scheme for RBSDEs Example: Starting and stopping problem Convergence for the obliquely RBSDE Representation using “switched” BSDEs “Switching” problem - “switched” BSDEs ◮ “Switching” strategy a = ( α j , θ j ) j starting at ( i , t ) N a = # { k ∈ N ∗ | θ k ≤ T } ◮ State process - cost process a s = α 0 1 0 ≤ s ≤ θ 0 + � N a s := � N a j =1 α j − 1 1 θ j − 1 < s ≤ θ j , A a j =1 c α j − 1 ,α j 1 θ j ≤ s ≤ T ◮ “Switched” BSDE (following the strategy a ) � T � T t = g a T ( X T ) + U a f a s ( X s , U a s , V a V a s d W s − A a T + A a s ) d s − t t t ◮ Representation ( a ∗ : optimal strategy) Y i a U a t = U a ∗ t = ess sup t J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Approximation of the forward SDE A Discretization scheme for RBSDEs Approximation of the RBSDE Convergence for the obliquely RBSDE Stability issue Approximation of the forward SDE ◮ For ( b , σ ) : R d → R d × M d Lipschitz : � t � t X t = X 0 + b ( X u ) d u + σ ( X u ) d W u 0 0 ◮ Euler scheme X with π = { 0 = t 0 < ... < t n < ... < t N = T } : � X π = X 0 0 X π X π t n + b ( X π t n )( t − t n ) + σ ( X π = t n )( W t − W t n ) , t ∈ ( t n , t n +1 ] t ◮ Error ( b , σ Lipschitz) � � 1 2 C E rr ( X , X π ) := E | X t − X π t | 2 sup ≤ √ N t ∈ [0 , T ] (max n | t n +1 − t n | ≤ C N ) J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Approximation of the forward SDE A Discretization scheme for RBSDEs Approximation of the RBSDE Convergence for the obliquely RBSDE Stability issue A scheme for the RBSDE: an example ◮ RBSDE, Snell envelop of l ( X t ): � T � T t ( Z u ) ′ d W u + Y t = l ( X T ) − t d K s , Y t ≥ l ( X t ) ◮ Discrete Snell envelop of ( l ( X π ) t n ) n : � � Y π � Y π t n := E t n +1 | F t n Y π t n := � Y π t n ∨ l ( X π t n ) → terminal condition Y π T := l ( X π ֒ T ). ◮ More general domain/reflection: Y π t n = � Y π t n ∨ l ( X π t n ) → Y π t n = P ( � Y π t n ) J-F Chassagneux Approximation of obliquely reflected BSDEs

Introduction Approximation of the forward SDE A Discretization scheme for RBSDEs Approximation of the RBSDE Convergence for the obliquely RBSDE Stability issue Moonwalk scheme for the RBSDE ◮ Implicit Euler scheme for the “BSDE part” : � � � t n , ¯ Y π Y π + ( t n +1 − t n ) f ( X π t n , Y π Z π t n := E t n +1 | F t n t n ) � � t n := ( t n +1 − t n ) − 1 E t n +1 ) ′ | F t n Z π ¯ ( W t n +1 − W t n )( Y π ◮ Taking into account the reflection Y π t n := � Y π ∈ℜ + P ( � Y π t n ) 1 t n ∈ℜ t n 1 t n / ℜ ⊂ π is the reflection grid with κ dates. T = � ◮ and terminal condition Y π Y π T := g ( X π T ). J-F Chassagneux Approximation of obliquely reflected BSDEs