Viscosity Solutions of Path-Dependent PDEs Zhenjie Ren CMAP, Ecole - PowerPoint PPT Presentation

Viscosity Solutions of Path-Dependent PDEs Zhenjie Ren CMAP, Ecole Polytechnique The 3rd young researchers meeting in Probability, Numerics and Finance June 29, 2016 Zhenjie Ren PPDE Le Mans, 29/06/2016 1 / 21

Motivation Table of Contents Motivation 1 From PDE to PPDE 2 Application in the control problems with delays 3 Zhenjie Ren PPDE Le Mans, 29/06/2016 2 / 21

Motivation PDE characterization : linear exmaple Linear Expectation � � � v ( t , x ) = E h ( W T ) � W t = x Zhenjie Ren PPDE Le Mans, 29/06/2016 3 / 21

Motivation PDE characterization : linear exmaple Linear Expectation Heat Equation � − ∂ t u − 1 2 D 2 � � v ( t , x ) = E h ( W T ) � W t = x x u = 0, u ( T , x ) = h ( x ) Zhenjie Ren PPDE Le Mans, 29/06/2016 3 / 21

Motivation PDE characterization : linear exmaple Linear Expectation Heat Equation � − ∂ t u − 1 2 D 2 � � v ( t , x ) = E h ( W T ) � W t = x x u = 0, u ( T , x ) = h ( x ) PDE characterization Function v is C 1 , 2 , and is a classical solution of the heat equation. Zhenjie Ren PPDE Le Mans, 29/06/2016 3 / 21

Motivation PDE characterization : linear exmaple Linear Expectation Heat Equation � − ∂ t u − 1 2 D 2 � � v ( t , x ) = E h ( W T ) � W t = x x u = 0, u ( T , x ) = h ( x ) PDE characterization Function v is C 1 , 2 , and is a classical solution of the heat equation. In the linear case, the martingale characterization as an alternative gives quite a lot analytic insight, and can be naturally generalized to the non-Markovian case. Zhenjie Ren PPDE Le Mans, 29/06/2016 3 / 21

Motivation PDE characterization : beyond the linear case Consider a controlled diffusion: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s , κ s ) ds + s , κ s ) dW s for κ ∈ K = { κ : κ t ∈ K for all t ∈ [0 , T ] } . Value function of optimal control � � T t f ( s , X κ s , κ s ) ds + h ( X κ � X κ � � v ( t , x ) = sup κ ∈K E T ) t = x Zhenjie Ren PPDE Le Mans, 29/06/2016 4 / 21

Motivation PDE characterization : beyond the linear case Consider a controlled diffusion: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s , κ s ) ds + s , κ s ) dW s for κ ∈ K = { κ : κ t ∈ K for all t ∈ [0 , T ] } . Value function of optimal control � � T t f ( s , X κ s , κ s ) ds + h ( X κ � � X κ � v ( t , x ) = sup κ ∈K E T ) t = x Hamilton-Jacobi-Bellman Equation b · Du + 1 ( σσ T ) D 2 u � � � � ∂ t u + sup k ∈ K 2 Tr + f = 0 , u ( T , x ) = h ( x ) . Zhenjie Ren PPDE Le Mans, 29/06/2016 4 / 21

Motivation PDE characterization : beyond the linear case Consider a controlled diffusion: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s , κ s ) ds + s , κ s ) dW s for κ ∈ K = { κ : κ t ∈ K for all t ∈ [0 , T ] } . Value function of optimal control � � T t f ( s , X κ s , κ s ) ds + h ( X κ � � X κ � v ( t , x ) = sup κ ∈K E T ) t = x Hamilton-Jacobi-Bellman Equation b · Du + 1 ( σσ T ) D 2 u � � � � ∂ t u + sup k ∈ K 2 Tr + f = 0 , u ( T , x ) = h ( x ) . PDE characterization (under some conditions) Function v is a viscosity solution of the HJB equation. Zhenjie Ren PPDE Le Mans, 29/06/2016 4 / 21

Motivation Non-Markovian model Consider the diffusion X controlled with delay: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s − δ , κ s ) ds + s − δ , κ s ) dW s , κ ∈ K Zhenjie Ren PPDE Le Mans, 29/06/2016 5 / 21

Motivation Non-Markovian model Consider the diffusion X controlled with delay: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s − δ , κ s ) ds + s − δ , κ s ) dW s , κ ∈ K Value function of optimal control � � T t f ( s , X κ s − δ , κ s ) ds + h ( X κ � � v t = sup κ ∈K E T ) � F t Zhenjie Ren PPDE Le Mans, 29/06/2016 5 / 21

Motivation Non-Markovian model Consider the diffusion X controlled with delay: � t � t X κ 0 b ( s , X κ 0 σ ( s , X κ t = X 0 + s − δ , κ s ) ds + s − δ , κ s ) dW s , κ ∈ K Value function of optimal control � � T t f ( s , X κ s − δ , κ s ) ds + h ( X κ � � v t = sup κ ∈K E T ) � F t It is IMPOSSIBLE to find a corresponding PDE of finite dimension state space ! Zhenjie Ren PPDE Le Mans, 29/06/2016 5 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian � � � v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian (Path-dependent) Heat Equation � � � − ∂ t u − 1 2 ∂ 2 v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) ωω u = 0, u ( T , ω ) = ξ ( ω ) Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian (Path-dependent) Heat Equation � � � − ∂ t u − 1 2 ∂ 2 v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) ωω u = 0, u ( T , ω ) = ξ ( ω ) How to make sense the equation (definition & existence/uniqueness)? Dupire derviatives, functional Itˆ o calculus ⇒ classical solution Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian (Path-dependent) Heat Equation � � � − ∂ t u − 1 2 ∂ 2 v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) ωω u = 0, u ( T , ω ) = ξ ( ω ) How to make sense the equation (definition & existence/uniqueness)? Dupire derviatives, functional Itˆ o calculus ⇒ classical solution Is there nonlinear extension ? Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian (Path-dependent) Heat Equation � � � − ∂ t u − 1 2 ∂ 2 v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) ωω u = 0, u ( T , ω ) = ξ ( ω ) How to make sense the equation (definition & existence/uniqueness)? Dupire derviatives, functional Itˆ o calculus ⇒ classical solution Is there nonlinear extension ? Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

Motivation A first meeting with Path-dependent PDE (PPDE) Linear Expectation: non-Markovian (Path-dependent) Heat Equation � � � − ∂ t u − 1 2 ∂ 2 v ( t , ω ) = E ξ ( W T ∧· ) � F t ( ω ) ωω u = 0, u ( T , ω ) = ξ ( ω ) How to make sense the equation (definition & existence/uniqueness)? Dupire derviatives, functional Itˆ o calculus ⇒ classical solution Is there nonlinear extension ? Introduce viscosity solutions to PPDE’s Zhenjie Ren PPDE Le Mans, 29/06/2016 6 / 21

From PDE to PPDE Table of Contents Motivation 1 From PDE to PPDE 2 Application in the control problems with delays 3 Zhenjie Ren PPDE Le Mans, 29/06/2016 7 / 21

From PDE to PPDE ‘The’ unique well-defined solution Consider the first order nonlinear equation with the boundary conditions: −| Du ( x ) | = − 1 , x ∈ ( − 1 , 1) , u ( − 1) = u (1) = 1 Zhenjie Ren PPDE Le Mans, 29/06/2016 8 / 21

From PDE to PPDE ‘The’ unique well-defined solution Consider the first order nonlinear equation with the boundary conditions: −| Du ( x ) | = − 1 , x ∈ ( − 1 , 1) , u ( − 1) = u (1) = 1 There is no smooth function, but infinite a.s. smooth functions satisfying this equation. Zhenjie Ren PPDE Le Mans, 29/06/2016 8 / 21

From PDE to PPDE ‘The’ unique well-defined solution Consider the first order nonlinear equation with the boundary conditions: −| Du ( x ) | = − 1 , x ∈ ( − 1 , 1) , u ( − 1) = u (1) = 1 There is no smooth function, but infinite a.s. smooth functions satisfying this equation. Is there a criteria which can select a unique solution? Zhenjie Ren PPDE Le Mans, 29/06/2016 8 / 21

From PDE to PPDE ‘The’ unique well-defined solution Consider the first order nonlinear equation with the boundary conditions: −| Du ( x ) | = − 1 , x ∈ ( − 1 , 1) , u ( − 1) = u (1) = 1 There is no smooth function, but infinite a.s. smooth functions satisfying this equation. Is there a criteria which can select a unique solution? Maximum Principle (Elliptic) max x ∈ O u ( x ) = max x ∈ ∂ O u ( x ), ∀ O ⊂ [ − 1 , 1] compact. Zhenjie Ren PPDE Le Mans, 29/06/2016 8 / 21

From PDE to PPDE ‘The’ unique well-defined solution Consider the first order nonlinear equation with the boundary conditions: −| Du ( x ) | = − 1 , x ∈ ( − 1 , 1) , u ( − 1) = u (1) = 1 There is no smooth function, but infinite a.s. smooth functions satisfying this equation. Is there a criteria which can select a unique solution? Maximum Principle (Elliptic) max x ∈ O u ( x ) = max x ∈ ∂ O u ( x ), ∀ O ⊂ [ − 1 , 1] compact. Only one continuous solution fits the maximum principle: u ( x ) = | x | . Zhenjie Ren PPDE Le Mans, 29/06/2016 8 / 21

From PDE to PPDE Why ‘the’ unique solution? Add a perturbation to the previous equation: −| Du ε ( x ) |− ε ∆ u ε = − 1 , x ∈ ( − 1 , 1) , u ε ( − 1) = u ε (1) = 1 Zhenjie Ren PPDE Le Mans, 29/06/2016 9 / 21