Stochastic target problems and pricing under risk constraints B. - PowerPoint PPT Presentation

The P − a.s. case ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) � � y : ∃ φ s.t. Z φ v ( t , x ) := inf t , x , y ∈ O on [ t , T ] . ✷ Theorem : For all φ and θ ∈ T [ t , T ] : GDP1 : Z φ t , z ∈ O on [ t , T ] ⇒ Y φ t , z ( θ ) ≥ v ( θ, X φ t , x ( θ )) GDP2 : � � Y φ t , z ( θ ) ≥ v ( θ, X φ t , x ( θ )) and Z φ y < v ( t , x ) ⇒ P t , z ∈ O on [ t , θ ] < 1 ✷ First introduced by Soner and Touzi for super-hedging under Gamma constraints. Extended to American type contraints : obstacle version of B. and Vu.

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p .

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p . ✷ Theorem : For all φ and θ ∈ T [ t , T ] : GDP1 : Z φ t , z ∈ O on [ t , T ] ⇒ Y φ t , z ( θ ) ≥ v ( θ, X φ t , x ( θ ) , p ) ?

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p . ✷ Theorem : For all φ and θ ∈ T [ t , T ] : GDP1 : Z φ t , z ∈ O on [ t , T ] ⇒ Y φ t , z ( θ ) ≥ v ( θ, X φ t , x ( θ ) , P t , p ( θ )) with � � G ( Z φ P t , p ( θ ) := E t , z ( T )) | F θ

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p . ✷ Theorem : For all φ and θ ∈ T [ t , T ] : GDP1 : Z φ t , z ∈ O on [ t , T ] ⇒ Y φ t , z ( θ ) ≥ v ( θ, X φ t , x ( θ ) , P t , p ( θ )) with � θ � � G ( Z φ P t , p ( θ ) := E t , z ( T )) | F θ = p + α s dW s , t if F t = σ ( W s , s ≤ t ) .

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p .

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p . ✷ Problem reduction : For all φ : � � Z φ G ( Z φ t , z ∈ O on [ t , T ] and E t , z ( T )) ≥ p if and only if ∃ α such that ( Z φ t , p ) ∈ O × R on [ t , T ] and G ( Z φ t , z , P α t , z ( T )) ≥ P α t , p ( T ) with � · � � G ( Z φ P t , p := E t , z ( T )) | F · = p + α s dW s . t

Constraints in expectations ✷ Problem extension : Z φ t , z = ( X φ t , x , Y φ t , x , y ) v ( t , x , p ) := � � � � y : ∃ φ s.t. Z φ G ( Z φ inf t , x , y ∈ O on [ t , T ] , E t , x , y ( T )) ≥ p . ✷ Problem reduction : For all φ : � � Z φ G ( Z φ t , z ∈ O on [ t , T ] and E t , z ( T )) ≥ p if and only if ∃ α such that ( Z φ t , p ) ∈ O × R on [ t , T ] and G ( Z φ t , z , P α t , z ( T )) ≥ P α t , p ( T ) with � · � � G ( Z φ P t , p := E t , z ( T )) | F · = p + α s dW s . t ✷ Can use the GDP with an increased controlled process.

PDE derivation

PDE derivation ✷ Previous works

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito.

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito. • B. : Jump diffusion with bounded control and locally bounded jumps. P − a.s. criteria.

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito. • B. : Jump diffusion with bounded control and locally bounded jumps. P − a.s. criteria. • B., Elie and Touzi : Brownian filtration with unbounded controls. Criteria in expectation (concentrating on the case of a criteria in expectation).

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito. • B. : Jump diffusion with bounded control and locally bounded jumps. P − a.s. criteria. • B., Elie and Touzi : Brownian filtration with unbounded controls. Criteria in expectation (concentrating on the case of a criteria in expectation). • B. and Vu : “American” case.

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito. • B. : Jump diffusion with bounded control and locally bounded jumps. P − a.s. criteria. • B., Elie and Touzi : Brownian filtration with unbounded controls. Criteria in expectation (concentrating on the case of a criteria in expectation). • B. and Vu : “American” case. • Moreau : Extension of B., Elie and Touzi to jump diffusions.

PDE derivation ✷ Previous works • Soner and Touzi : Brownian filtration and bounded controls (apart from particular cases in finance). P − a.s. criteria. Problems with Gamma constraints with Zhang, Cheridito. • B. : Jump diffusion with bounded control and locally bounded jumps. P − a.s. criteria. • B., Elie and Touzi : Brownian filtration with unbounded controls. Criteria in expectation (concentrating on the case of a criteria in expectation). • B. and Vu : “American” case. • Moreau : Extension of B., Elie and Touzi to jump diffusions. ✷ In the following, we consider the case with controls of bounded variations types (simplification of a work with M. N. Dang).

The general model ✷ Set of controls : L ∈ L set of continuous non-decreasing R d -valued adapted processes L s.t. E � | L | 2 � < ∞ . T

The general model ✷ Set of controls : L ∈ L set of continuous non-decreasing R d -valued adapted processes L s.t. E � | L | 2 � < ∞ . T ✷ Dynamics of Z = ( X , Y ) ∈ R d × R : dX L = µ X ( X L ) dr + σ X ( X L ) dW + β X ( X L ) dL dY L = µ Y ( Z L ) dr + σ Y ( Z L ) dW + β Y ( Z L ) dL .

The general model ✷ Set of controls : L ∈ L set of continuous non-decreasing R d -valued adapted processes L s.t. E � | L | 2 � < ∞ . T ✷ Dynamics of Z = ( X , Y ) ∈ R d × R : dX L = µ X ( X L ) dr + σ X ( X L ) dW + β X ( X L ) dL dY L = µ Y ( Z L ) dr + σ Y ( Z L ) dW + β Y ( Z L ) dL . ✷ Problem : y : ∃ L ∈ L / Z L G ( Z L � � � � v ( t , x , p ) := inf t , x , y ∈ O , E t , x , y ( T )) ≥ p

The general model ✷ Set of controls : L ∈ L set of continuous non-decreasing R d -valued adapted processes L s.t. E � | L | 2 � < ∞ . T ✷ Dynamics of Z = ( X , Y ) ∈ R d × R : dX L = µ X ( X L ) dr + σ X ( X L ) dW + β X ( X L ) dL dY L = µ Y ( Z L ) dr + σ Y ( Z L ) dW + β Y ( Z L ) dL . ✷ Problem : y : ∃ L ∈ L / Z L G ( Z L � � � � v ( t , x , p ) := inf t , x , y ∈ O , E t , x , y ( T )) ≥ p ✷ Reduction : A set of predictable square integrable processes y : ∃ ( L , α ) ∈ L × A / Z L t , x , y ∈ O , G ( Z L t , x , y ( T )) ≥ P α � � inf t , p ( T ) .

Formal derivation of the PDE Assume that v is smooth and the inf is achieved. For y = v ( t , x , p ) , ∃ ( L , α ) such that Z L t , z ∈ O on [ t , T ] and G ( Z L t , x , y ( T )) ≥ P α t , p ( T ) .

Formal derivation of the PDE Assume that v is smooth and the inf is achieved. For y = v ( t , x , p ) , ∃ ( L , α ) such that Z L t , z ∈ O on [ t , T ] and G ( Z L t , x , y ( T )) ≥ P α t , p ( T ) . Then Y L t , z ( t +) ≥ v ( t + , X L t , x ( t +) , P α t , p ( t +)) and

Formal derivation of the PDE Assume that v is smooth and the inf is achieved. For y = v ( t , x , p ) , ∃ ( L , α ) such that Z L t , z ∈ O on [ t , T ] and G ( Z L t , x , y ( T )) ≥ P α t , p ( T ) . Then Y L t , z ( t +) ≥ v ( t + , X L t , x ( t +) , P α t , p ( t +)) and µ Y ( z ) − L α � � X , P v ( t , x , p ) dt ≥ ( σ Y ( z ) − D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α t ) dW t + ( β Y ( z ) − D x v ( t , x , p ) β X ( x )) dL t

Formal derivation of the PDE µ Y ( z ) − L α � � X , P v ( t , x , p ) dt ≥ ( σ Y ( z ) − D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α t ) dW t + ( β Y ( z ) − D x v ( t , x , p ) β X ( x )) dL t

Formal derivation of the PDE µ Y ( z ) − L α � � X , P v ( t , x , p ) dt ≥ ( σ Y ( z ) − D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α t ) dW t + ( β Y ( z ) − D x v ( t , x , p ) β X ( x )) dL t µ Y ( x , v ( t , x , p )) − L α Ok if X , P v ( t , x , p ) ≥ 0 with σ Y ( x , v ( t , x , p )) = D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α .

Formal derivation of the PDE µ Y ( z ) − L α � � X , P v ( t , x , p ) dt ≥ ( σ Y ( z ) − D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α t ) dW t + ( β Y ( z ) − D x v ( t , x , p ) β X ( x )) dL t µ Y ( x , v ( t , x , p )) − L α Ok if X , P v ( t , x , p ) ≥ 0 with σ Y ( x , v ( t , x , p )) = D x v ( t , x , p ) σ X ( x ) − D p v ( t , x , p ) α . Or ( β Y ( x , v ( t , x , p )) − D x v ( t , x , p ) β X ( x )) ℓ > 0 with ℓ ∈ ∆ + := ∂ B 1 ( 0 ) ∩ R d + .

Formal derivation of the PDE Set µ Y ( · , v ) − L α � � Fv := sup X , P v , α ∈ Nv Gv := max { [ β Y ( · , v ) − D x v ( t , x ) β X ( x )] ℓ, ℓ ∈ ∆ + } with Nv := { α : σ Y ( · , v ) = D x v σ X + D p v α } R d ∆ + := + ∩ ∂ B 1 ( 0 ) . PDE characterization in the interior of the domain max { Fv , Gv } = 0 on ( t , x , v ( t , x )) ∈ int ( D ) where D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } .

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } .

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } . Assumption : D ∈ C 1 , 2 (or intersection of C 1 , 2 domains).

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } . Assumption : D ∈ C 1 , 2 (or intersection of C 1 , 2 domains). Take δ ∈ C 1 , 2 such that δ > 0 in int ( D ) , δ = 0 on ∂ D and δ < 0 elsewhere.

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } . Assumption : D ∈ C 1 , 2 (or intersection of C 1 , 2 domains). Take δ ∈ C 1 , 2 such that δ > 0 in int ( D ) , δ = 0 on ∂ D and δ < 0 elsewhere. The state constraints imposes d δ ( t , Z L t , z ( t )) ≥ 0 if ( t , z ) ∈ ∂ D .

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } . Assumption : D ∈ C 1 , 2 (or intersection of C 1 , 2 domains). Take δ ∈ C 1 , 2 such that δ > 0 in int ( D ) , δ = 0 on ∂ D and δ < 0 elsewhere. The state constraints imposes d δ ( t , Z L t , z ( t )) ≥ 0 if ( t , z ) ∈ ∂ D . As above it implies : either L Z δ ( t , x , y ) ≥ 0 and D δ ( t , x , y ) σ Z ( x , y ) = 0

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) Domain is D := { ( t , x , y ) : ( x , y ) ∈ O ( t ) } . Assumption : D ∈ C 1 , 2 (or intersection of C 1 , 2 domains). Take δ ∈ C 1 , 2 such that δ > 0 in int ( D ) , δ = 0 on ∂ D and δ < 0 elsewhere. The state constraints imposes d δ ( t , Z L t , z ( t )) ≥ 0 if ( t , z ) ∈ ∂ D . As above it implies : or max { D δ ( t , x , y ) β z ( x , y ) ℓ, ℓ ∈ ∆ + } > 0 .

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) The GDP and the need for a reflexion on the boundary leads to the definition of N in v := { α ∈ Nv : D δ ( · , v ) σ Z ( · , v ) = 0 } F in v µ Y ( · , v ) − L α � � := sup min X , P v , L Z δ ( · , v ) α ∈ N in v G in v := ℓ ∈ ∆ + min { [ β Y ( · , v ) − D x v β X ] ℓ , D δ ( · , v ) β z ( · , v ) ℓ } max

PDE on the space boundary ( x , y ) ∈ ∂ O ( t ) The GDP and the need for a reflexion on the boundary leads to the definition of N in v := { α ∈ Nv : D δ ( · , v ) σ Z ( · , v ) = 0 } F in v µ Y ( · , v ) − L α � � := sup min X , P v , L Z δ ( · , v ) α ∈ N in v G in v := ℓ ∈ ∆ + min { [ β Y ( · , v ) − D x v β X ] ℓ , D δ ( · , v ) β z ( · , v ) ℓ } max Then, the PDE on the boundary reads max { F in 0 v , G in v } = 0 on ( t , x , v ( t , x )) ∈ ∂ D .

Example Pricing of the WVAP-guaranteed liquidation contract

The VWAP guaranted pricing problem ✷ K stocks to liquidate.

The VWAP guaranted pricing problem ✷ K stocks to liquidate. ✷ Has an impact on prices

The VWAP guaranted pricing problem ✷ K stocks to liquidate. ✷ Has an impact on prices ✷ Ensure that will guarantee a mean selling price of γ × the mean selling price of the market.

The VWAP guaranted pricing problem ✷ K stocks to liquidate. ✷ Has an impact on prices ✷ Ensure that will guarantee a mean selling price of γ × the mean selling price of the market. ✷ What is the price of the guarantee ?

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks.

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t ✷ Cumulated gain from liquidation : dY L = X L , 1 dL t

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t ✷ Cumulated gain from liquidation : dY L = X L , 1 dL t ✷ Volume weighted market price : dX L , 2 = X L , 1 d ϑ .

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t ✷ Cumulated gain from liquidation : dY L = X L , 1 dL t ✷ Volume weighted market price : dX L , 2 = X L , 1 d ϑ . ✷ Cumulated # of sold stocks : X L , 3 := L ∈ [Λ , ¯ Λ] → { K }

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t ✷ Cumulated gain from liquidation : dY L = X L , 1 dL t ✷ Volume weighted market price : dX L , 2 = X L , 1 d ϑ . ✷ Cumulated # of sold stocks : X L , 3 := L ∈ [Λ , ¯ Λ] → { K } ✷ Risk constraint (with γ ∈ ( 0 , 1 ) ) � � �� X L , 3 t , x , y ( T ) − K γ X L , 2 Y L t , x ∈ [Λ , Λ] and E ℓ t , x ( T ) ≥ p } .

The VWAP guaranted pricing problem ✷ Controls : L ↑ adapted and continuous. L t = # of sold stocks. ✷ Price dynamics : dX L , 1 = X L , 1 µ ( X L , 1 ) dt + X L , 1 σ ( X L , 1 ) dW t − X L , 1 β ( X L , 1 ( t )) dL t ✷ Cumulated gain from liquidation : dY L = X L , 1 dL t ✷ Volume weighted market price : dX L , 2 = X L , 1 d ϑ . ✷ Cumulated # of sold stocks : X L , 3 := L ∈ [Λ , ¯ Λ] → { K } ✷ Pricing function (with Ψ( x , y ) = ℓ ( y − γ Kx 2 ) , γ > 0) � � v ( t , x , p ) := inf { y ≥ 0 : ∃ L s.t. X L , 3 Ψ( Z L t , x ∈ [Λ , Λ] , E t , x , y ( T )) ≥ p } .

PDE characterization Proposition Under “good assumptions”, v ∗ is a viscosity supersolution on [ 0 , T ) of F ϕ , x 1 + x 1 β D x 1 ϕ − D x 3 ϕ = 0 if Λ ≤ x 3 ≤ Λ � � max and v ∗ is a subsolution on [ 0 , T ) of F ϕ , x 1 + x 1 β D x 1 ϕ − D x 3 ϕ Λ < x 3 < Λ � � �� min ϕ , max = 0 if ϕ , x 1 + β D x 1 ϕ − D x 3 ϕ � � Λ = x 3 min = 0 if x 3 = Λ , min { ϕ , F ϕ } = 0 if where F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 Moreover, v ∗ ( T , x , p ) = v ∗ ( T , x , p ) = Ψ − 1 ( x , p ) .

PDE characterization Proposition Under “good assumptions”, v ∗ is a viscosity supersolution on [ 0 , T ) of F ϕ , x 1 + x 1 β D x 1 ϕ − D x 3 ϕ = 0 if Λ ≤ x 3 ≤ Λ � � max and v ∗ is a subsolution on [ 0 , T ) of F ϕ , x 1 + x 1 β D x 1 ϕ − D x 3 ϕ Λ < x 3 < Λ � � �� min ϕ , max = 0 if ϕ , x 1 + β D x 1 ϕ − D x 3 ϕ � � Λ = x 3 min = 0 if x 3 = Λ , min { ϕ , F ϕ } = 0 if where F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 Moreover, v ∗ ( T , x , p ) = v ∗ ( T , x , p ) = Ψ − 1 ( x , p ) .

The “good assumptions” ✷ On Λ , Λ : Λ , Λ ∈ C 1 , Λ < ¯ Λ on [ 0 , T ) , D Λ , D Λ ∈ ( 0 , M ]

The “good assumptions” ✷ On Λ , Λ : Λ , Λ ∈ C 1 , Λ < ¯ Λ on [ 0 , T ) , D Λ , D Λ ∈ ( 0 , M ] ✷ On the loss function ℓ : ∃ ǫ > 0 s.t. ǫ ≤ D − ℓ , D + ℓ ≤ ǫ − 1 , r →∞ D + ℓ ( r ) = lim r →∞ D − ℓ ( r ) . and lim

Control on the gradients ✷ Proposition v ∗ is a viscosity supersolution of min { D p ϕ − ǫ , ( D x 1 ϕ − CD p ϕ ) 1 x 1 > 0 , − D x 1 ϕ + CD p ϕ } = 0 ( ∗ ) and v ∗ is a viscosity subsolution of max {− D p ϕ + ǫ , ( D x 1 ϕ − CD p ϕ ) 1 x 1 > 0 , − D x 1 ϕ + CD p ϕ } = 0 . ( ∗∗ ) where C is continuous and depends only on x .

Control on the gradients ✷ Proposition v ∗ is a viscosity supersolution of min { D p ϕ − ǫ , ( D x 1 ϕ − CD p ϕ ) 1 x 1 > 0 , − D x 1 ϕ + CD p ϕ } = 0 ( ∗ ) and v ∗ is a viscosity subsolution of max {− D p ϕ + ǫ , ( D x 1 ϕ − CD p ϕ ) 1 x 1 > 0 , − D x 1 ϕ + CD p ϕ } = 0 . ( ∗∗ ) where C is continuous and depends only on x . ✷ Provides a control on the ratio D x 1 ϕ/ D p ϕ in F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2

More controls on v

More controls on v ✷ It also implies that ∃ η > 0 s.t. 0 ≤ v ( t , x , p ) ≤ ǫ − 1 | p − ℓ ( 0 ) | + γη ( 1 + | x | ) ,

More controls on v ✷ It also implies that ∃ η > 0 s.t. 0 ≤ v ( t , x , p ) ≤ ǫ − 1 | p − ℓ ( 0 ) | + γη ( 1 + | x | ) , ✷ and that for ( t n , x n , p n ) n s.t. ( t n , x n ) → ( t , x ) : n →∞ v ∗ ( t n , x n , p n ) = 0 if p n → −∞ , n →∞ v ∗ ( t n , x n , p n ) = lim lim v ∗ ( t n , x n , p n ) v ∗ ( t n , x n , p n ) 1 lim = lim = D ℓ ( ∞ ) if p n → ∞ . p n p n n →∞ n →∞

More controls on v ✷ It also implies that ∃ η > 0 s.t. 0 ≤ v ( t , x , p ) ≤ ǫ − 1 | p − ℓ ( 0 ) | + γη ( 1 + | x | ) , ✷ and that for ( t n , x n , p n ) n s.t. ( t n , x n ) → ( t , x ) : n →∞ v ∗ ( t n , x n , p n ) = 0 if p n → −∞ , n →∞ v ∗ ( t n , x n , p n ) = lim lim v ∗ ( t n , x n , p n ) v ∗ ( t n , x n , p n ) 1 lim = lim = D ℓ ( ∞ ) if p n → ∞ . p n p n n →∞ n →∞ ✷ A little more : v is continuous in p and x 3 .

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions.

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions. ✷ Recall that F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions. ✷ Recall that F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 ✷ We now control D x 1 ϕ/ D p ϕ . This is not enough... If we need to penalize in x 1 (stock price) then the therm | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ will blow up as n → ∞ , where n 2 | 2 du to the doubling of comes from the usual penalisation n | x 1 1 − x 1 constants.

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions. ✷ Recall that F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 ✷ We now control D x 1 ϕ/ D p ϕ . This is not enough... If we need to penalize in x 1 (stock price) then the term | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ will blow up as n → ∞ , where n comes 2 | 2 due to the doubling of from the usual penalisation n | x 1 1 − x 1 constants.

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions. ✷ Recall that F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 ✷ We now control D x 1 ϕ/ D p ϕ . Assumption : x 1 > 0 s.t. µ (ˆ x 1 ) ≤ 0 = σ (ˆ x 1 ) . ∃ ˆ

Uniqueness ✷ Want a comparison resul in the class of function with the above limit and growth conditions. ✷ Recall that F ϕ := −L X ϕ − ( x 1 σ ) 2 � � | D x 1 ϕ/ D p ϕ | 2 D 2 p ϕ − 2 ( D x 1 ϕ/ D p ϕ ) D 2 ( x 1 , p ) ϕ . 2 ✷ We now control D x 1 ϕ/ D p ϕ . Assumption : x 1 > 0 s.t. µ (ˆ x 1 ) ≤ 0 = σ (ˆ x 1 ) . ∃ ˆ ✷ Bound on the stock price...

Comparison ✷ Theorem : Let U (resp. V ) be a non-negative super- and subsolutions which are continuous in x 3 . Assume that U ( t , x , p ) ≥ V ( t , x , p ) if t = T or x 1 ∈ { 0 , 2 ˆ x 1 } , and that ∃ c + > 0 and c − ∈ R s.t. V ( t ′ , x ′ , p ′ ) / p ′ ≤ c + ≤ ( t ′ , y ′ , p ′ ) → ( t , y , ∞ ) U ( t ′ , y ′ , p ′ ) / p ′ , lim sup lim inf ( t ′ , x ′ , p ′ ) → ( t , x , ∞ ) V ( t ′ , x ′ , p ′ ) ≤ c − ≤ ( t ′ , y ′ , p ′ ) → ( t , y , −∞ ) U ( t ′ , y ′ , p ′ ) . lim sup lim inf ( t ′ , x ′ , p ′ ) → ( t , x , −∞ ) If either U is a supersolution of (*) which is continuous in p , or V is a subsolution of (**) which is continuous in p , then U ≥ V .

Additional remarks