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

1

Motivation

2

Solution space Example of the semilinear equations Solution space and its pseudometric

3

Derivatives Dupire’s derivative and C 2 Piecewise differentiable: C

2 4

Capacities 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 Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

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:

Yt = ξ(BHQ) + ˆ HQ

t∧HQ

F(Bs, Ys, Zs)ds − ˆ HQ

t∧HQ

ZsdBs. (1) 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 Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

Example of semilinear PDE

To connect the semilinear PDE (2) and the BSDE (1), Darling and Pardoux used the following group of BSDEs: u(x) := Y x

0 = ξ(Bx Hx

Q)+

ˆ Hx

Q

F(Bx

s , Y x s , Z x s )ds−

ˆ Hx

Q

Z x

s dBx s . (3)

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 Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

Semilinear PPDE

The semilinear PPDE should be in the form: −1 2∂2

ωωu − F(ω, u, ∂ωu) = 0 in Q, u = ξ on ∂Q.

(4) The corresponding BSDE (non-Markov): Yt = ξ(BHQ∧·) + ˆ HQ

t∧HQ

F(Bs∧·, Ys, Zs)ds − ˆ HQ

t∧HQ

ZsdBs. (5) Intuitively, we need define u by the following group of BSDEs: u(ω) := Y ω

0 = ξ(Bω Hx

Q∧·)+

ˆ Hω

Q

F(Bω

s∧·, Y ω s , Z ω s )ds −

ˆ Hω

Q

Z ω

s dBω s .

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: Ωe.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

Solution space

Ωe := {ω ∈ Ω : ∃t ≥ 0, ω = ωt∧·} , The Dirichlet problem is set on the domain Q := {ω ∈ Ωe : ωt ∈ Q, ∀t ∈ R+} ,where Q is an open, bounded and convex subset of Rd. Definition We define the following pseudometric: de(ω, ˜ ω) := inf

l∈I

sup

t∈[0,+∞)

|ωl(t) − ˜ ωt|, ∀ω, ˜ ω ∈ Ωe where I is the set of all the increasing bijections from R+ to R+. The regularity under de(·, ·) ensures us the ellipticity of the solution, i.e., ∂tu = 0

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

Why de(·, ·) ?

Example We consider a special case of BSDE, where F = 0, ξ(ω) = ˆ ¯

t(ω)

ω(s)ds, H(ω) = inf {t : |ω(t)| > 1} . Notice that ξ is not continuous w.r.t. de(·, ·). Then, u(t, ω) = Y t,ω(0) = E [ξ ((ω ⊕t B)H∧·)] . If ωt∧· > 1, that is, H(ω) ≤ t, E [ξ ((ω ⊕t B)H∧·)] = ˆ t ωsds + E ˆ Ht,ω (ωt + Bs)ds

• ,

the t-derivative is equal to ωt.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Example of the semilinear equations Solution space and its pseudometric

Property (P)

Indeed, we have the following sufficient condition. Proposition If F and ξ satisfy the following property (P) : let ϕ : Ωe → R, (P) For all ω ∈ Ω and ω

′ ∈ Ωe: ϕ(ω ⊗s ω ′) = ϕ(ω ⊗s 0 ⊗s+h ω ′),

∀s, h ≥ 0 . Then we deduce that ∂tu = 0. The property P is closely related to the distance de(·, ·). Proposition Let ϕ be defined on Q. The following two statements are equivalent: (1) ϕ is continuous w.r.t. the distance de(·, ·), 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 Solution space Derivatives Capacities Dupire’s derivative and C2 Piecewise differentiable: C2

Derivative and C 2

Following the idea of Ekren, Touzi and Zhang [2, 3], we define the differentiability by Itˆ

• ’s formula.

Definition We say u ∈ C 2(Ωe), if u ∈ C(Ωe) and there exist ∂ωu ∈ C

• Ωe; Rd

, ∂2

ωωu ∈ C

• Ωe; Sd

such that, for any ω ∈ Ω and any P ∈ P∞, {u(ωt∧·)}t≥0 is a local P-semimartingale and it holds: du(ωt∧·) = ∂ωu(ωt∧·) · dBt + 1 2∂2

ωωu(ωt∧·) : d Bt , P − a.s. (6)

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities Dupire’s derivative and C2 Piecewise differentiable: C2

Piecewise differentiable: C

2

Definition

Let u : Q → R. We say u ∈ C

2(Q), if u is bounded, u(ωt∧·) is continuous in t,

and there exists an increasing sequence of F-stopping times {Hn; n ≥ 1}, such that: (1) for each i and ω, HHi ,ω

i+1

∈ HωHi ∧· whenever Hi(ω) < HQ(ω) < ∞; the set {i : Hi(ω) < HQ(ω)} is finite P∞

0 -q.s. ω; and limi→∞ CL 0 [Hω i < Hω Q ] = 0;

(2) for each i there exist ∂ωui, ∂2

ωωui such that for all ω,

• ∂ωuiωHi ∧· ,
• ∂2

ωωuiωHi ∧· are continuous on Oω i :=

• ω

′

t∧· ∈ Ωe : t ≤ HHi ,ω i+1

• and that for all t ∈ [Hi, Hi+1):

u(ωt∧·) − u(ωHi ∧·) = ˆ t

Hi

∂ωui(ωs∧·) · dBs + ˆ t

Hi

1 2∂2

ωωui(ωs∧·) : d Bs , P∞

− q.s.

We will construct the piecewise differentiable super- and sub-solutions to the PPDEs.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities The families of probabilities The optimal stopping theorem

Capacities

We define the families of probabilities: for L > 0, PL,0 := {P : ∃α ∈ Bd

L, β ∈ BS 0, √ 2L, and W P a Browian motion,

Bt = ˆ t αsds + ˆ t βsdW P

s , P − a.s.},

PL,1 := {P : ∃β ∈ BS √

2/L, √ 2L, and W P a Browian motion,

Bt = ˆ t βsdW P

s , P − a.s.}.

Then define the capacities and the nonlinear expectations: for i = 0, 1, CL,i [·] := sup

P∈PL,i P [·] , E L,i [·] := sup P∈PL,i EP [·] , EL,i [·] :=

inf

P∈PL,i EP [·] .

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Motivation Solution space Derivatives Capacities The families of probabilities The optimal stopping theorem

Optimal stopping

For X ∈ L0(Λ) and ∀(t, ω) such that ωt∧· ∈ D,

S

L t [XHD∧·] (ω) := sup τ∈T t E L,0 t

• (Xτ∧HD)t,ω

.

SL

t can be defined in a contrary way.

Theorem Let µ > 0, HD ∈ H and X ∈ BUC(D). Define the Sneil’s envelop:

Y := S

L

e−µ(HD∧·)XHD∧·

• ,

and τ ∗ := inf {t ≥ 0 : Yt = e−µtXt} . Then Yτ ∗ = e−µτ ∗Xτ ∗. Y is an E

L,0-supermartingale on [0, HD], and an E L,0-martingale on

[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

5

Definition of the viscosity solutions Viscosity solutions Assumption for the well-posedness

6

Consistency with the classic solution

7

Uniqueness Ideas of proving the uniqueness Constructive proof

8

Existence

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Viscosity solutions Assumption for the well-posedness

Fully nonlinear elliptic PPDE

The fully nonlineqr elliptic PPDE has the form as:

• Lu(ω) := −G(ω, u, ∂ωu, ∂2

ωωu) = 0,

ω ∈ Q; u = ξ, ω ∈ ∂Q. (7) For any u ∈ BUC(Q), ω ∈ Q, and L > 0, µ > 0, define the set of the test functions:

AL,µu(ω) :=

• ϕ ∈ C 2(D) : (ϕ − uω)0 = 0 = SL

e−µ(Hω

D ∧·)(ϕ − uω)Hω D ∧·

• ,

A

L,µu(ω) :=

• ϕ ∈ C 2(D) : (ϕ − uω)0 = 0 = S

L

e−µ(Hω

D ∧·)(ϕ − uω)Hω D ∧·

• .

Notice that if we only consider the viscosity solutions to the uniformly elliptic equations, then the exponential discount is unnecessary.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Viscosity solutions Assumption for the well-posedness

Viscosity solutions

Definition (i) Let L > 0. We say u ∈ BUC(Q) is a viscosity (L, µ)-subsolution (resp. (L, µ)-supersolution) of PPDE (7) if, for ω ∈ Q and any ϕ ∈ AL,µu(ω) (resp. ϕ ∈ A

L,µu(ω)):

−G(ω, ϕ(0), ∂ωϕ(0), ∂2

ωωϕ(0)) ≤ (resp. ≥) 0.

(ii) We say u ∈ BUC(Q) is a viscosity subsolution (resp. supersolution) of PPDE (7) if u is a viscosity (L, µ)-subsolution (resp. (L, µ)-supersolution) of PPDE (7) for some L > 0 and µ > 0. (iii) We say u ∈ BUC(Q) is a viscosity solution of PPDE (7) if it is both a viscosity subsolution and a viscosity supersolution.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Viscosity solutions Assumption for the well-posedness

Assumption for the well-posedness

Assumption (Structural)

|G(·, 0, z, 0)| ≤ C0 uniformly in z. G is uniformly elliptic, i.e., there exists L0 > 0 such that for γ1 ≥ γ2 G(ω, y, z, γ1) − G(ω, y, z, γ2) ≥ 1 L0 Id : (γ1 − γ2). Confined on Ωe, G is uniformly continuous in ω under de(·, ·). And G is uniformly Lipschitz continuous in (y, z, γ) with a Lipschitz constant L0. G is strictly decreasing in y, i.e., there exists µ0 > 0 such that for y2 ≥ y1 G(ω, y1, z, γ) − G(ω, y2, z, γ) ≥ µ0(y2 − y1). the boundary condition h is uniformly continuous.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Consistency with the classic solution

Proposition (From v.s. to c.s.) If u ∈ C 2(Q) is a viscosity solution to the PPDE (7), then u is a classic solution. Proposition (From c.s. to v.s.) If u ∈ C 2(Q) is a classic solution to the PPDE(7), then u is also a viscosity solution.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Consistency with the classic solution

Proof. (1) v.s.⇒c.s. is trivial. We can take u itself as the test function. (2) c.s.⇒v.s. We consider a simple equation: −G(∆u) = 0. Suppose that u is not a viscosity subsolution. Then, there exists a test function ϕ so that 0 = (u − ϕ)0 = max

τ

E

L,0

e−µτ∧H(u − ϕ)τ∧H

• ,

and −G(∆ϕ0) > 0. Since u and ϕ are both C 2, we deduce that ∆(u − ϕ)0 ≤ 0. Therefore, we have −G(∆u0) > 0. It is a contradiction against the fact that u is classic solution.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Partial comparison principle

We intend to show the comparison principle. Theorem Comparison principleAssume that u1 (resp. u2) is a viscosity subsolution (resp. supersolution) to the PPDE (7). If u1 ≤ u2 on ∂Q, then u1 ≤ u2 in Q. To reach this, we first prove a partial comparison principle. Theorem Partial comparison principleLet u2 ∈ BUC(Q) be a viscosity supersolution of PPDE (7) and u1 ∈ C

2(Q) be bounded and

satisfying Lu1(ω) ≤ 0. ∀ω ∈ Q. If u1 ≤ u2 on ∂Q, then u1 ≤ u2 in Q.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Comparison principle

Set the following two functions:

u(ω) := inf

• ψ(ω) : ψ ∈ D

ξ Q(ω)

• , u(ω) := sup
• ψ(ω) : ψ ∈ Dξ

Q(ω)

• ,

where

D

ξ Q(ω)

:=

• ψ ∈ C

2(Qω) : ψ is bounded, Lωψ ≥ 0 in Q, ψ ≥ ξω on ∂Q

• ,

Dξ

Q(ω)

:=

• ψ ∈ C

2(Qω) : ψ is bounded, Lωψ ≤ 0 in Q, ψ ≤ ξω on ∂Q

• .

Proposition Partial comparison principle + ’u = u’ ⇒ Comparison principle (Uniqueness).

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Construction by path-frozen PDEs

We want to show u = u. The strategy is to construct a function θ, such that θ + ǫ ∈ D

ξ Q and θ − ǫ ∈ Dξ Q.

As the beginning, let us treat the semilinear equation. In this case, we can first verify that the solution to the BSDE, u(ωt∧·) = Yt = ξ(BHQ∧·)+ ˆ HQ

t∧HQ

F(Bs∧·, Ys, Zs)ds − ˆ HQ

t∧HQ

ZsdBs, is the viscosity solution to the PPDE. It remains to construct a piecewisely smooth θ such that θ + ǫ ≥ u and θ − ǫ ≤ u.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Path-frozen PDEs

Given a nonlinearity G, define the following function on R × Rd × Sd: gω(y, z, γ) := G(ω, y, z, γ), ω ∈ Q. For any ǫ > 0, set Oǫ :=

• x ∈ Rd : |x| < ǫ
• . Denote

Oq

ǫ := Oǫ ∩ Q. Also we denote by Oq,ω ǫ

for all ω ∈ Q: Oq,ω

ǫ

:= Oǫ ∩ Qω¯

t(ω).

The path-frozen PDE with the nonlinearity gω is: (E)ω

ǫ

Lωv := −gω(v, Dv, D2v) = 0 in Oq,ω

ǫ

. Hence, when ω is fixed, it is simply a PDE.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Construction for the semilinear case

Denote a sequence of stopping times

H0 = 0, Hi+1 := inf{t ≥ Hi : |Bt − BHi | ≥ ǫ} ∧ HQ.

Denote by a ⊖ b the interpolation. Especially, we note πn := ⊖0≤i≤nBHi. Define a new BSDE, so that on (Hi, Hi+1] we have

Y ǫ

t = Y ǫ Hi+1 +

ˆ Hi+1

t

F(πi, Y ǫ

s , Z ǫ s )ds −

ˆ Hi+1

t

Z ǫ

s dBs.

(8)

Notice that the difference between the solution to the modified BSDE and u is well controlled (by comparison). Moreover, the equation (8) is indeed Markov, so related to a PDE:

−1 2∂2

xxθi − F(πi, θi, ∂xθi)

= in Oq

ǫ ,

θi(x; πi) = θi+1(0; πi ⊖ x) on ∂Oq

ǫ .

So given (t, ω) and t ∈ (Hi, Hi+1], define θ(ωt∧·) := θi(ωt − ωHi; πi). This θ is what we want.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Construction for the fully nonlinear case

The same idea in the fully nonlinear context leads us to show the following proposition. Proposition (Path-frozen PDE) There exists a sequence of continuous functions (πn, x) → θǫ

n(πn, x), bounded uniformly in (ǫ, n), and such that

θǫ

n(πn; ·) is the viscosity solution of:

Lπnθǫ

n := −gπn(θǫ n, Dθǫ n, D2θǫ n) = 0 in Oq,πn ǫ

. and

• θǫ

n(πn; x) = ξ(ωπx

n),

|x| < ǫ and x ∈ ∂Qπn, θǫ

n(πn; x) = θǫ n+1(πx n; 0),

|x| = ǫ. Notice that θ(ωt∧·) := θǫ

i (ωt − ωHi; πi) has not been yet

piecewisely smooth. To realize the piecewise regularisation, we

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence Ideas of proving the uniqueness Constructive proof

Assumption on the path-frozen PDEs

Assumption (Technical) For ǫ > 0, ω ∈ Q, and h ∈ C(∂Oq,ω

ǫ

), we have v = v, where v(x) := inf{w(x) : w ∈ C 2(Oq,ω

ǫ

) ∩ C(cl(Oq,ω

ǫ

)), Lωw ≥ 0 in Oq,ω

ǫ

, w ≥ h on ∂Oq,ω

ǫ

}, v(x) := sup{w(x) : w ∈ C 2(Oq,ω

ǫ

) ∩ C(cl(Oq,ω

ǫ

)), Lωw ≤ 0 in Oq,ω

ǫ

, w ≤ h on ∂Oq,ω

ǫ

}. This assumption is strong and not trivial to verify it. However, it is the common default of Perron’s approach.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Existence of the solutions

During the proof of the uniquess, we have already constructed a function u = u = u. Intuitively, if the solution to the PPDE exists, it must be u. To verify it, we have two things to show:

1 u is bounded and uniformly continuous. 2 u satisfies the viscosity property.

The former one results from the uniform continuity of θǫ (the solutions to the path-frozen PDEs), and the later one follows from a similar argument of the consistency of the solution.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Thank you!

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Ekren, Touzi and Zhang, On viscosity solution to semilinear PPDEs. Ekren, Touzi and Zhang, Viscosity soltuions of fully nonlinear parabolic path dependent PDEs: Part I. Ekren, Touzi and Zhang, Viscosity soltuions of fully nonlinear parabolic path dependent PDEs: Part II. Ekren, Touzi and Zhang, Optimal stopping under nonlinear expectation. Darling and Pardoux, BSDEs with random terminal. Briand and Hu, Stability of BSDEs with random terminal time and homogemzation of semilinear elliptic PDEs. Nutz and Van Handel, Constructing Sublinear Expectations on Path Space Stochastic Processes and their Applications.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs

Definition of the viscosity solutions Consistency with the classic solution Uniqueness Existence

Gilbarg, Trudinger, Elliptic partial differential equations of second order.

REN Zhenjie Fully Nonlinear Elliptic Path Dependent PDEs