Switching problems and related BSDE approximation Romuald ELIE - PowerPoint PPT Presentation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Switching problems and related BSDE approximation Romuald ELIE CEREMADE, Université Paris-Dauphine Joint work with J.-F. Chassagneux & I. Kharroubi Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Outline of the talk Starting and stopping problem (d=2) Numerical resolution of BSDE Numerical resolution of BSDE with oblique reflections An alternative approach : Constrained BSDEs with jumps Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps Starting and Stopping problem Hamadene & Jeanblanc 05 : Consider e.g. a power station producing electricity whose price is given by a diffusion process X : dX t = b ( t , X t ) dt + σ ( t , X t ) dW t Two modes for the power station : mode 1 : operating, with running profit f 1 ( X t ) dt and terminal one g 1 ( X T ) mode 0 : closed, with running profit f 0 ( X t ) dt and terminal one g 0 ( X T ) ֒ → switching from one mode to another has a cost : c > 0 Management decides 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, i.e. the management strategy. Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps Value processes Following a strategy a from t up to T , gives Z T X J ( a , t ) = g a T ( X T ) + f a s ( X s ) ds − c 1 { t ≤ θ j ≤ T } t j ≥ 0 The value processes starting respectively at time 0 in mode 1 and 2 are Y 0 Y 1 0 := sup E [ J ( a , 0 )] and 0 := sup E [ J ( a , 0 )] { a ∈A s . t . a 0 = 0 } { a ∈A s . t . a 0 = 1 } Y is solution of a coupled optimal stopping problem »Z τ – Y 0 f 0 ( X s ) ds + ( Y 1 t = ess sup τ − c ) 1 { τ< T } | F t E t ≤ τ ≤ T t »Z τ – Y 1 f 1 ( X s ) ds + ( Y 0 t = ess sup E τ − c ) 1 { τ< T } | F t t ≤ τ ≤ T t with terminal conditions : Y 0 T = g 0 ( X T ) and Y 1 T = g 1 ( X T ) The optimal strategy ( θ ∗ j , α ∗ j ) is given by α ∗ α ∗ α ∗ j + 1 := 1 − α ∗ θ ∗ j + 1 := inf { s ≥ θ ∗ j j + 1 and j | Y = Y − c } s s j Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps System of reflected BSDEs Y is the solution of the following system of reflected BSDEs : Z T Z T Z T Y i Z i dK i t = g i ( X T ) + f i ( X s ) ds − s · dW s + s , i ∈ { 0 , 1 } , t t t with (the coupling...) Y 1 t ≥ Y 0 t − c and Y 0 t ≥ Y 1 t − c , ∀ t ∈ [ 0 , T ] and (‘optimality’ of K ) Z T Z T “ ” “ ” Y 1 s − ( Y 0 dK 1 Y 0 s − ( Y 1 dK 0 s − c ) s = 0 and s − c ) s = 0 0 0 • Problem : Oblique reflections. • Idea : Interpret Y 1 − Y 0 as the solution of a doubly reflected BSDE. Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps Related PDE Associated coupled system of PDE on R × [ 0 , T ) “ ” min − ∂ t u 0 − L u 0 − f 0 , u 0 − u 1 + c = 0 “ ” min − ∂ t u 1 − L u 1 − f 1 , u 1 − u 0 + c = 0 L : u �→ σ 2 with 2 ∂ xx u + b ∂ x u Terminal conditions u 0 ( T , . ) = g 0 ( . ) and u 1 ( T , . ) = g 1 ( . ) Link via Y 0 Y 1 t = u 0 ( t , X t ) and t = u 1 ( t , X t ) Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps Non exhaustive Literature Literature on optimal switching : Hamadène & Jeanblanc 05 : starting and stopping problem ( d = 2). Djehiche, Hamadène & Popier 07 : studied the multidimentional case. Carmona & Ludkovski 06 or Porchet, Touzi & Warin 07 : Additional constraints and numerical results. Link with non linear Backward SDE : Hu & Tang 07 “multi-dimentional BSDEs with oblique reflection” BSDE representation for optimal switching in the case where X uncontrolled or h i at most partially controlled : dX a t = σ ( X a µ a ( X a t ) t ) dt + dW t . Hamadène & Zhang 08 Generalization of Hu & Tang’s BSDEs but still with an uncontrolled underlying diffusion. Literature on control : Bouchard 09 : Relation with stochastic target problems with jumps. Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Representation using reflected BSDEs Obliquely reflected BSDEs Constrained BSDE with jumps Multi-dimensional reflected BSDE Multi-dimensional reflected BSDE (see Hamadène & Zhang 08) : Find m triplets ( Y i , Z i , K i ) i ∈I ∈ ( S 2 × L 2 ( W ) × A 2 ) I satisfying 8 R T R T > Y i t f i ( s , X s , Y 1 s , . . . , Y m s , Z i t Z i s dW s + K i T − K i t = g i ( X T ) + s ) ds − > < t t ≥ h i , j ( t , Y j Y i t ) > R T > : 0 [ Y i t − max j ∈I { h i , j ( t , Y j t ) } ] dK i t = 0 Conditions on the constraint h in order to avoid instantaneous gain via circle switching. For any i � = j , h i , j and f i are increasing in y j . = ⇒ ’Interpretation’ in terms of cooperative game options The reflections are oblique with respect to the domain of definition of Y . Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps FBSDE system Z t Z t 8 > • FSDE > X t = x + b ( s , X s ) ds + σ ( s , X s ) dW s > > < 0 0 > Z T Z T > > > • BSDE : Y t = g ( X T ) + f ( s , X s , Y s , Z s ) ds − Z s dW s t t Solution and link with PDE (Pardoux & Peng, 90 & 92) ; » – 1 »Z 1 – 1 2 2 | Y r | 2 | Z r | 2 dr � Y � S 2 := E sup < ∞ , � Z � L 2 := E < ∞ , 0 ≤ r ≤ 1 0 PDE L X [ y ] + f ( ., y , σ ∇ y ) = 0 y ( T , . ) = g ( . ) Approximation of the BM (Chevance 97, Briand 01, Ma 02) ; Discrete time scheme based on the path regularity of Z (Zhang) ; Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Discrete time scheme (Zhang 02) Z t Z t 8 > • FSDE > X t = x + b ( s , X s ) ds + σ ( s , X s ) dW s > > < 0 0 > Z T Z T > > > : • BSDE Y t = g ( X T ) + f ( s , X s , Y s , Z s ) ds − Z s dW s t t • Regular time grid π := ( t i ) i ≤ n on [ 0 , T ] • Forward Euler approximation X π of X X π Initial value : 0 := x X π t i + 1 := X π t i + 1 n b ( t i , X π t i ) + σ ( t i , X π From t i to t i + 1 : t i )( W t i + 1 − W t i ) • Backward approximation ( Y π , Z π ) of ( Y , Z ) Y π T := g ( X π Terminal value : T ) 8 ˆ ˜ Z π Y π := n E t i + 1 ( W t i + 1 − W t i ) | F t i > < t i From t i + 1 to t i : > ˆ ˜ ` ´ : Y π Y π + 1 t i , X π t i , Y π t i , Z π := t i + 1 | F t i n f E t i t i Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Intuition of the scheme Z t i + 1 Z t i + 1 Y t i = Y t i + 1 + f ( r , X r , Y r , Z r ) dr − Z r · dW r t i t i Z π given by the representation of Y π Step 1 : Constant step driver ( e t i + 1 ) Z t i + 1 ` ´ t i + 1 + 1 Y π t i = Y π t i , X π t i , Y π t i , Z π Z π e n f − r · dW r t i t i Z π by F t i -meas. r.v. Step 2 : Best L 2 (Ω × [ t i , t i + 1 ]) approximation of e "Z t i + 1 # ˆ ˜ Z π e Z π Y π t i := n E r dr | F t i = n E t i + 1 ( W t i + 1 − W t i ) | F t i t i Step 3 : Conditioning the first expression ˆ ˜ ` ´ + 1 Y π Y π t i , X π t i , Y π t i , Z π t i = E t i + 1 | F t i n f . t i Romuald ELIE Switching problems and related BSDE approximation

Starting and Stopping problem Numerical resolution of BSDE Obliquely reflected BSDEs Constrained BSDE with jumps Approximation Error (Zhang 02) Y t = y ( t , X t ) L X [ y ] + f ( ., y , σ ∇ y ) = 0 • PDE y ( 1 , . ) = g ( . ) • Forward Euler approximation X π of X X π X π t i + 1 := X π t i + 1 n b ( t i , X π t i ) + σ ( t i , X π 0 := x and t i )( W t i + 1 − W t i ) • Backward approximation ( Y π , Z π ) of ( Y , Z ) 8 ˆ ˜ Z π Y π := n E t i + 1 ( W t i + 1 − W t i ) | F t i > < t i Y π T := g ( X π T ) & > ˆ ˜ ` ´ : Y π Y π + 1 t i , X π t i , Y π t i , Z π := E t i + 1 | F t i n f t i t i • Approximation Error X n ˆ t i | 2 ˜ ˆ t i | 2 ˜ E rr ( Z , Z π ) := 1 E rr ( Y , Y π ) := sup | Y t i − Y π | Z t i − Z π E E n t i i = 1 E rr ( Y , Y π ) + E rr ( Z , Z π ) ≤ C | π | Romuald ELIE Switching problems and related BSDE approximation