Fully Nonlinear Elliptic Path Dependent PDEs The viscosity solutions - PowerPoint PPT Presentation

Fully Nonlinear Elliptic Path Dependent PDEs The viscosity solutions to the Dirichlet problem REN Zhenjie Ecole Polytechnique, Paris Joint work with Nizar Touzi September, 2013 Journ´ ee DIM RDM-IdF 2013 REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Introduction and Preliminaries Motivation 1 Solution space 2 Example of the semilinear equations Solution space and its pseudometric Derivatives 3 Dupire’s derivative and C 2 2 Piecewise differentiable: C Capacities 4 The families of probabilities The optimal stopping theorem REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Motivation Driven by the nonlinear and non-Markov modeling the control theory, the well-posedness of the fully nonlinear path dependent PDEs (PPDEs) need to be concerned systematically. I. Ekren, N. Touzi and J. Zhang [1, 2, 3] have recently proposed an approach to the well-posedness of the viscosity solutions to the fully nonlinear parabolic PPDEs. It is natural to extend the research to the elliptic case. REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Example of semilinear PDE Darling and Pardoux [5] have studied the relation between the solutions to the Markov BSDEs with random terminal and the viscosity solutions to the Dirichlet problems of the semilinear PDEs. They studied the BSDE: ˆ H Q ˆ H Q Y t = ξ ( B H Q ) + F ( B s , Y s , Z s ) ds − Z s dB s . (1) t ∧ H Q t ∧ H Q The corresponding PDE is: − 1 2∆ u − F ( x , u , Du ) = 0 in Q , u = ξ on ∂ Q . (2) REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Example of semilinear PDE To connect the semilinear PDE (2) and the BSDE (1), Darling and Pardoux used the following group of BSDEs: ˆ H x ˆ H x Q Q u ( x ) := Y x 0 = ξ ( B x F ( B x s , Y x s , Z x Z x s dB x Q )+ s ) ds − s . (3) H x 0 0 x is the initial value of the diffusion B . Proposition (Darling and Pardoux) Under some general assumptions, u defined as in (3) is the viscosity solution to the Dirichlet problem of the semilinear PDE (2). We are interested in the same result in the path-dependent context. REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Semilinear PPDE The semilinear PPDE should be in the form: − 1 2 ∂ 2 ωω u − F ( ω, u , ∂ ω u ) = 0 in Q , u = ξ on ∂ Q . (4) The corresponding BSDE (non-Markov): ˆ H Q ˆ H Q Y t = ξ ( B H Q ∧· ) + F ( B s ∧· , Y s , Z s ) ds − Z s dB s . (5) t ∧ H Q t ∧ H Q Intuitively, we need define u by the following group of BSDEs: ˆ H ω ˆ H ω Q Q u ( ω ) := Y ω 0 = ξ ( B ω F ( B ω s ∧· , Y ω s , Z ω Z ω s dB ω Q ∧· )+ s ) ds − s . H x 0 0 Therefore, the solution u should be defined at the ω for which we can concatenate the canonical paths afterwards: ω ⊗ B . This observation leads to our proposal of the solution space: Ω e . REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Solution space Ω e := { ω ∈ Ω : ∃ t ≥ 0 , ω = ω t ∧· } , The Dirichlet problem is set on the domain Q := { ω ∈ Ω e : ω t ∈ Q , ∀ t ∈ R + } , where Q is an open, bounded and convex subset of R d . Definition We define the following pseudometric: d e ( ω, ˜ ω ∈ Ω e ω ) := inf sup | ω l ( t ) − ˜ ω t | , ∀ ω, ˜ l ∈I t ∈ [0 , + ∞ ) where I is the set of all the increasing bijections from R + to R + . The regularity under d e ( · , · ) ensures us the ellipticity of the solution, i.e., ∂ t u = 0 REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Why d e ( · , · ) ? Example We consider a special case of BSDE, where ˆ ¯ t ( ω ) F = 0 , ξ ( ω ) = ω ( s ) ds , H ( ω ) = inf { t : | ω ( t ) | > 1 } . 0 Notice that ξ is not continuous w.r.t. d e ( · , · ). Then, u ( t , ω ) = Y t ,ω (0) = E [ ξ (( ω ⊕ t B ) H ∧· )] . If � ω t ∧· � > 1, that is, H ( ω ) ≤ t , ˆ t � ˆ H t ,ω � E [ ξ (( ω ⊕ t B ) H ∧· )] = ω s ds + E ( ω t + B s ) ds , 0 0 the t -derivative is equal to ω t . REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Example of the semilinear equations Derivatives Solution space and its pseudometric Capacities Property (P) Indeed, we have the following sufficient condition. Proposition If F and ξ satisfy the following property (P) : let ϕ : Ω e → R , ′ ∈ Ω e : ϕ ( ω ⊗ s ω ′ ) = ϕ ( ω ⊗ s 0 ⊗ s + h ω ′ ) , (P) For all ω ∈ Ω and ω ∀ s , h ≥ 0 . Then we deduce that ∂ t u = 0. The property P is closely related to the distance d e ( · , · ). Proposition Let ϕ be defined on Q . The following two statements are equivalent: (1) ϕ is continuous w.r.t. the distance d e ( · , · ), q.s.; (2) ϕ satisfies the property (P) and is continuous w.r.t. the supremum norm �·� q.s.. REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Dupire’s derivative and C 2 Solution space Piecewise differentiable: C 2 Derivatives Capacities Derivative and C 2 Following the idea of Ekren, Touzi and Zhang [2, 3], we define the differentiability by Itˆ o’s formula. Definition We say u ∈ C 2 (Ω e ), if u ∈ C (Ω e ) and there exist � Ω e ; R d � , ∂ 2 � Ω e ; S d � ∂ ω u ∈ C ωω u ∈ C such that, for any ω ∈ Ω and any P ∈ P ∞ , { u ( ω t ∧· ) } t ≥ 0 is a local P -semimartingale and it holds: du ( ω t ∧· ) = ∂ ω u ( ω t ∧· ) · dB t + 1 2 ∂ 2 ωω u ( ω t ∧· ) : d � B � t , P − a . s . (6) REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Dupire’s derivative and C 2 Solution space Piecewise differentiable: C 2 Derivatives Capacities 2 Piecewise differentiable: C Definition 2 ( Q ), if u is bounded, u ( ω t ∧· ) is continuous in t , Let u : Q → R . We say u ∈ C and there exists an increasing sequence of F -stopping times { H n ; n ≥ 1 } , such that: (1) for each i and ω , H H i ,ω ∈ H ω Hi ∧· whenever H i ( ω ) < H Q ( ω ) < ∞ ; the set i +1 0 -q.s. ω ; and lim i →∞ C L 0 [ H ω i < H ω { i : H i ( ω ) < H Q ( ω ) } is finite P ∞ Q ] = 0; ωω u i such that for all ω , (2) for each i there exist ∂ ω u i , ∂ 2 ∂ ω u i � ω Hi ∧· , ωω u i � ω Hi ∧· are continuous on O ω � t ∧· ∈ Ω e : t ≤ H H i ,ω � ′ ∂ 2 � � i := ω i +1 and that for all t ∈ [ H i , H i +1 ): ˆ t ∂ ω u i ( ω s ∧· ) · dB s u ( ω t ∧· ) − u ( ω H i ∧· ) = H i ˆ t 1 2 ∂ 2 ωω u i ( ω s ∧· ) : d � B � s , P ∞ + − q . s . 0 H i We will construct the piecewise differentiable super- and sub-solutions to the PPDEs. REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space The families of probabilities Derivatives The optimal stopping theorem Capacities Capacities We define the families of probabilities: for L > 0, 2 L , and W P a Browian motion , P L , 0 { P : ∃ α ∈ B d L , β ∈ B S := √ 0 , ˆ t ˆ t β s dW P B t = α s ds + s , P − a . s . } , 0 0 2 L , and W P a Browian motion , P L , 1 { P : ∃ β ∈ B S √ := √ 2 / L , ˆ t β s dW P B t = s , P − a . s . } . 0 Then define the capacities and the nonlinear expectations: for i = 0 , 1, L , i [ · ] := sup C L , i [ · ] := sup P ∈P L , i E P [ · ] , E L , i [ · ] := P ∈P L , i E P [ · ] . P ∈P L , i P [ · ] , E inf REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space The families of probabilities Derivatives The optimal stopping theorem Capacities Optimal stopping For X ∈ L 0 (Λ) and ∀ ( t , ω ) such that ω t ∧· ∈ D , L L , 0 ( X τ ∧ H D ) t ,ω � � S t [ X H D ∧· ] ( ω ) := sup τ ∈T t E . t S L t can be defined in a contrary way. Theorem Let µ > 0 , H D ∈ H and X ∈ BUC ( D ) . Define the Sneil’s envelop: L � � e − µ ( H D ∧· ) X H D ∧· Y := S , and τ ∗ := inf { t ≥ 0 : Y t = e − µ t X t } . Then Y τ ∗ = e − µτ ∗ X τ ∗ . Y is L , 0 -supermartingale on [0 , H D ] , and an E L , 0 -martingale on an E [0 , τ ∗ ] . Consequently, τ ∗ is an optimal stopping time. REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Fully nonlinear elliptic PPDE Definition of the viscosity solutions 5 Viscosity solutions Assumption for the well-posedness Consistency with the classic solution 6 Uniqueness 7 Ideas of proving the uniqueness Constructive proof Existence 8 REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs