Robust feedback switching control Erhan BAYRAKTAR University of - PowerPoint PPT Presentation

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Robust feedback switching control Erhan BAYRAKTAR ∗ ∗ University of Michigan Based on joint works with Huyen PHAM, University Paris Diderot, LPMA Andrea COSSO, University Paris Diderot ICERM, Brown University, June 22, 2017 Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Switching control • Switching control : sequence of interventions ( ι n ) n that occur at random times ( τ n ) n due to switching costs, and naturally arises in investment problems with fixed transaction costs or in real options. • Standard approach : open-loop ( � = closed-loop) control give the evolution for the controlled state process, with assigned drift and diffusion coefficients. • In practice, the coefficients are obtained through estimation procedures and are unlikely to coincide with the real coefficients. • Robust approach : switching control problem robust to a misspecification of the model for the controlled state process. Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Robust/Game formulation • We formulate the problem as a game : switcher vs nature (model uncertainty). ◮ We consider the two-step optimization problem � � sup inf υ J ( α, υ ) . α • What definition for the switching control α and for υ ? Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Feedback formulation • Elliott-Kalton formulation (Fleming-Souganidis 89) : α non-anticipative strategy and υ open-loop control , i.e. the switcher knows the current and past choices made by nature In practice, the switcher only knows the evolution of the state process. ◮ Feedback formulation α feedback switching control ( closed-loop control ) = ⇒ feedback formulation of the switching control problem. υ open-loop control (nature is aware of the all information at disposal) ↔ Knightian uncertainty → zero-sum control/control game but not symmetric Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Outline Model setup 1 Stochastic Perron’s method and Hamilton-Jacobi-Bellman-Isaacs 2 equation Ergodicity 3 Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Outline Model setup 1 Stochastic Perron’s method and Hamilton-Jacobi-Bellman-Isaacs 2 equation Ergodicity 3 Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Robust feedback switching system • Fixed (Ω , F , P ), T > 0, and W a d -dimensional Brownian motion. For any ( s , x , i ) ∈ [0 , T ] × R d × I m , consider the system on R d × I m , with I m = { 1 , . . . , m } the set of regimes :  � t � t X t = x + s b ( X r , I r , υ r ) dr + s σ ( X r , I r , υ r ) dW r , s � t � T ,     I t = i 1 { s � t <τ 0 ( X · , I ·− ) } + �  n ∈ N ι n ( X · , I · − )1 { τ n ( X · , I ·− ) � t <τ n +1 ( X · , I ·− ) } , s � t < T ,    I s − = I s , I T = I T − . ◮ υ : [ s , T ] × Ω → U is an open-loop control adapted to a filtration F s = ( F s t ) t � s satisfying the usual conditions. U compact metric space. U s , s : class of all open-loop controls starting at s . Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Feedback switching controls L ([ s , T ]; I m ) space of c` agl` ad paths valued in I m . B s = ( B s t ) t ∈ [ s , T ] natural filtration of C ([ s , T ]; R d ) × L ([ s , T ]; I m ). T s family of all B s -stopping times valued in [ s , T ]. ◮ Feedback switching control α = ( τ n , ι n ) n ∈ N where : Switching times : τ n ∈ T s and s � τ 0 � · · · � τ n � · · · � T . Interventions : ι n : C ([ s , T ]; R d ) × L ([ s , T ]; I m ) → I m is B s τ n -measurable, for any n ∈ N . ◮ A s , s : class of all feedback switching controls starting at s . Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Existence and uniqueness result (H1) b and σ jointly continuous on R d × I m × U and | b ( x , i , u ) − b ( x ′ , i , u ) | + � σ ( x , i , u ) − σ ( x ′ , i , u ) � � L | x − x ′ | . Proposition Let (H1) hold. Then, for every ( s , x , i ) ∈ [0 , T ] × R d × I m , α ∈ A s , s , υ ∈ U s , s , there exists a unique F s -adapted solution ( X s , x , i ; α, u , I s , x , i ; α, u ) t ∈ [ s , T ] to the feedback system, satisfying : t t Every path of ( X s , x , i ; α,υ , I s , x , i ; α,υ ) belongs to · · − C ([ s , T ]; R d ) × L ([ s , T ]; I m ) . For any p � 1 there exists a positive constant C p , T such that � | p � | X s , x , i ; α,υ � C p , T (1 + | x | p ) . E sup t t ∈ [ s , T ] Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Value function of robust switching control problem Feedback control/open-loop control game : ∀ ( s , x , i ) ∈ [0 , T ] × R d × I m , V ( s , x , i ) := sup υ ∈U s , s J ( s , x , i ; α, υ ) , inf α ∈A s , s with � � T f ( X s , x , i ; α,υ , I s , x , i ; α,υ J ( s , x , i ; α, υ ) := E , υ r ) dr r r s + g ( X s , x , i ; α,υ , I s , x , i ; α,υ ) T T � � c ( X s , x , i ; α,υ , I s , x , i ; α,υ , I s , x , i ; α,υ − )1 { s � τ n < T } , τ n τ n τ − n n ∈ N where τ n stands for τ n ( X s , x , i ; α,υ , I s , x , i ; α,υ ). · · − Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation  � � � − ∂ V  ∂ t ( s , x , i ) − inf u ∈ U L i , u V ( s , x , i ) + f ( x , i , u ) min ,   �� � [0 , T ) × R d × I m V ( s , x , i ) − max j � = i V ( s , x , j ) − c ( x , i , j ) = 0 ,    ( x , i ) ∈ R d × I m , V ( T , x , i ) = g ( x , i ) , where � � L i , u V ( s , x , i ) = b ( x , i , u ) . D x V ( s , x , i ) + 1 ⊺ ( x , i , u ) D 2 2tr σσ x V ( s , x , i ) . ◮ First aim : prove that V is a viscosity solution to the dynamic programming HJBI equation : by stochastic Perron method : avoiding the direct proof of Dynamic Programming Principle (DPP) Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Outline Model setup 1 Stochastic Perron’s method and Hamilton-Jacobi-Bellman-Isaacs 2 equation Ergodicity 3 Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Stochastic Perron : main idea Developed in a series of papers by B. and Sirbu • Define stochastic sub and super-solutions as functions that satisfy (roughly) half of the DPP ◮ with these definitions, sub and super-solutions envelope the value function • Consider sup of sub-solutions and inf of super-solutions (Perron) : v − := sup of sub-solutions � V � v + := inf of super-solutions ◮ Show that v − is a viscosity super-solution and v + is a viscosity sub-solution. • Comparison principle → v − = V = v + is the unique continuous viscosity solution . and (as a byproduct) V satisfies the DPP Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Some comments • Stochastic semi-solutions have to be carefully defined (depending on the control problem) → constructive proof for the existence of a viscosity solution comparing with the value function linear, control, optimal stopping problems (Bayraktar-Sirbu, 12, 13, 14.) Erhan BAYRAKTAR Robust feedback switching control

Introduction Model setup Stochastic Perron’s method and HJBI equation Ergodicity Conclusion Stochastic semisolutions Definition (Stochastic subsolutions V − ) v stochastic subsolution to the HJBI equation if : v is continuous, v ( T , x , i ) � g ( x , i ) for any ( x , i ) ∈ R d × I m , and | v ( s , x , i ) | sup ( s , x , i ) ∈ [0 , T ] × R d × I m < ∞ , for some q � 1. 1+ | x | q Half-DPP property. For any s ∈ [0 , T ] and τ, ρ ∈ T s with ι n ) n ∈ N ∈ A s ,τ + such that, for any τ � ρ � T , there exists � α = ( � τ n , � α = ( τ n , ι n ) n ∈ N ∈ A s , s , υ ∈ U s , s , and ( x , i ) ∈ R d × I m , we have � � ρ ′ v ( τ ′ , X τ ′ , I τ ′ ) � E τ ′ f ( X t , I t , υ t ) dt + v ( ρ ′ , X ρ ′ , I ρ ′ ) � � � � � � F s − c ( X � n , I ( � n ) − , I � n )1 { τ ′ � � τ ′ τ ′ τ ′ τ ′ n <ρ ′ } τ ′ n ∈ N with the shorthands X = X s , x , i ; α ⊗ τ � α,υ , I = I s , x , i ; α ⊗ τ � α,υ . ◮ The set of stochastic supersolutions V + is defined similarly. Erhan BAYRAKTAR Robust feedback switching control