Viscosity Solutions of Fully Nonlinear Path Dependent PDEs

Motivation and examplesMain results in the parabolic case

Elliptic path-dependent PDEs

Viscosity Solutions of Fully Nonlinear PathDependent PDEs

Nizar TOUZI

Ecole Polytechnique, France

Collaborators:

Ibrahim EKREN , Zhen-Jie REN, and Jianfeng ZHANG

Hammamet, October 14, 2013

Nizar TOUZI Viscosity Solutions of PPDEs



Parabolic nonlinear PPDEsLinear equationsViscosity solutions of second order elliptic PDEs

Outline

1 Motivation and examplesParabolic nonlinear PPDEsLinear equationsViscosity solutions of second order elliptic PDEs

2 Main results in the parabolic caseConsistency, stability, partial comparison

3 Elliptic path-dependent PDEsFormulationA purely probabilistic comparison result





Notations

• Ω = ω ∈ C 0([0,T ],Rd), ω0 = 0, ‖ω‖ = supt≤T |ωt |

• Λ = [0,T ]× Ω, d[(t, ω), (t ′, ω′)

]= |t − t ′|+ ‖ω.∧t − ω′.∧t′‖

• B canonical process, i.e. Bt(ω) = ω(t)

• F = Ft the corresponding filtration, i.e. Ft = σ(Bs , s ≤ t)

• P0 : Wiener measure on Ω, and for α, β, F−predictable(appropriate dimension and integrability)

Pα,β := P0 (∫ .

0αtdt + βtdBt

)−1

• PL :=Pα,β : |α| ≤ L, |β|2 ≤ 2L





Our objective : nonlinear path-dependent PDEs

Find prog. meas. process u(t, ω) satisfying the parabolic equation :− ∂tu − G (., u, ∂ωu, ∂

2ωωu)

(t, ω) = 0, t < T , ω ∈ Ω,

u(T , ω) = ξ(ω)

where ξ(ω) = ξ((ωs)s≤T

)and G (t, ω, y , z , γ) is F−prog. meas.

G : [0,T ]× Ω× R× Rd × Sd −→ R

Note : prog. meas. =⇒

u(t, ω) = u(t, (ωs)s≤t

)G (t, ω, y , z , γ) = G

(t, (ωs)s≤t , y , z , γ

)Nizar TOUZI Viscosity Solutions of PPDEs




Differentiability of processes

• For ϕ ∈ C 0(Λ), the right time-derivative is defined by Dupire :

∂tϕ(t, ω) := limh→0,h>0

1h

[ϕ(t + h, ω·∧t

)− ϕ

(t, ω)], t < T

whenever limit exists

Definition ϕ ∈ C 1,2(Λ) if ϕ ∈ C 0(Λ), ∂tϕ ∈ C 0(Λ), and thereexist Z ∈ C 0(Λ,Rd), Γ ∈ C 0(Λ, Sd) s.t.

dϕt = ∂tϕtdt + ZtdBt +12

Γt : d 〈B 〉t, P-a.s. for all P ∈ ∪L>0PL

Denote ∂ωϕ := Z and ∂2ωωϕ := Γ





Differentiability of processes

• For ϕ ∈ C 0(Λ), the right time-derivative is defined by Dupire :

∂tϕ(t, ω) := limh→0,h>0

1h

[ϕ(t + h, ω·∧t

)− ϕ

(t, ω)], t < T

whenever limit exists

Definition ϕ ∈ C 1,2(Λ) if ϕ ∈ C 0(Λ), ∂tϕ ∈ C 0(Λ), and thereexist Z ∈ C 0(Λ,Rd), Γ ∈ C 0(Λ, Sd) s.t.

dϕt = ∂tϕtdt + ZtdBt +12

Γt : d 〈B 〉t, P-a.s. for all P ∈ ∪L>0PL

Denote ∂ωϕ := Z and ∂2ωωϕ := Γ





Path-dependent heat equation : the smooth case

• By using the r.c.p.d. define for ξ ∈ L1(P0) :

u(t, ω) := EPt,ω0[ξ]

for all t ≤ T , ω ∈ Ω

• Assume that u ∈ C 1,2, then :

dut =(∂tut +

12∂2ωωut

)dt + ∂ωutdBt , P0 − a.s.

Since u is a P0−martingale, we obtain the heat equation :

∂tu +12∂2ωωu = 0 and uT = ξ

• Note ut(ω) := EPt,ω0[BT

2

]is not C 1,2










dut =(∂tut +

12∂2ωωut





2

]is not C 1,2










dut =(∂tut +

12∂2ωωut





2

]is not C 1,2





More examples I

• Backward SDE (Pardoux & Peng ’91...) :

dYt = −Ft(ω,Yt ,Zt)dt + ZtdBt , YT = ξ, P0 − a.s.

if u(t, ω) := Yt(ω) is C 1,2, then u solves the semilinear P-PDE

−∂tu −12∂2ωωu − F (., u, ∂ωu) = 0

• Second order backward SDE (Cheridito, Soner, T., Victoir ’06,Soner, T. & Zhang ’12) =⇒ fully nonlinear P-PDE

Existing literature can be viewed as a theory of Sobolev solutions ofP-PDE





More examples II

• Stochastic control of non-Markov systems :

dX νt = b

(t,X ν

., νt)dt + σ

(t,X ν

., νt)dBt

and

u(t, x.) = supν∈U

E[ ∫ T

tL(s,X ν

., νs)ds + gT (X ν

.)]

=⇒ Path-dependent HJB equation

• and the corresponding stochastic differential games =⇒Path-dependent HJBI equation

• In particular, control problems with delay





Standard viscosity solutions [M. Crandal & P.-L. Lions ’83]

g(x , y , z , γ) nondecreasing in γ. Consider the PDE :

(E) − g(., v ,Dv ,D2v)(x) = 0, x ∈ O (open subset of Rd)

Exercise For v ∈ C 2(O), the following are equivalent :(i) v is a supersolution of (E)(ii) For all (x0, φ) ∈ O × C 2(O) :

(φ− v)(x0) = maxO

(φ− v) =⇒ −g(., v ,Dφ,D2φ)(x0) ≥ 0





Standard viscosity solutions [M. Crandal & P.-L. Lions ’83]

g(x , y , z , γ) nondecreasing in γ. Consider the PDE :

(E) − g(., v ,Dv ,D2v)(x) = 0, x ∈ O (open subset of Rd)

Exercise For v ∈ C 2(O), the following are equivalent :(i) v is a supersolution of (E)(ii) For all (x0, φ) ∈ O × C 2(O) :

(φ− v)(x0) = maxO

(φ− v) =⇒ −g(., v ,Dφ,D2φ)(x0) ≥ 0





Standard viscosity solutions : consistency

Assume (ii), then the first and second order conditions for(φ− v)(x0) = maxO(φ− v) are

D(φ− v)(x0) = 0 and D2(φ− v)(x0) ≤ 0

Since v is a classical supersolution :

0 ≤ −g(., v ,Dv ,D2v)(x0)

= −g(., v ,Dφ,D2φ)(x0)

+(gz(...)D(φ− v)︸︷︷︸

=0

+ gγ(...)︸︷︷︸≥0

D2(φ− v)︸︷︷︸≤0

)(x0)

≤ −g(., v ,Dφ,D2φ)(x0)







D(φ− v)(x0) = 0 and D2(φ− v)(x0) ≤ 0


0 ≤ −g(., v ,Dv ,D2v)(x0)

= −g(., v ,Dφ,D2φ)(x0)

+(gz(...)D(φ− v)︸︷︷︸

=0

+ gγ(...)︸︷︷︸≥0

D2(φ− v)︸︷︷︸≤0

)(x0)

≤ −g(., v ,Dφ,D2φ)(x0)







D(φ− v)(x0) = 0 and D2(φ− v)(x0) ≤ 0


0 ≤ −g(., v ,Dv ,D2v)(x0)

= −g(., v ,Dφ,D2φ)(x0)

+(gz(...)D(φ− v)︸︷︷︸

=0

+ gγ(...)︸︷︷︸≥0

D2(φ− v)︸︷︷︸≤0

)(x0)

≤ −g(., v ,Dφ,D2φ)(x0)







D(φ− v)(x0) = 0 and D2(φ− v)(x0) ≤ 0


0 ≤ −g(., v ,Dv ,D2v)(x0)

= −g(., v ,Dφ,D2φ)(x0)

+(gz(...)D(φ− v)︸︷︷︸

=0

+ fγ(...)︸︷︷︸≥0

D2(φ− v)︸︷︷︸≤0

)(x0)

≤ −g(., v ,Dφ,D2φ)(x0)





Standard definition of viscosity solutions

Let

Av(x) :=ϕ ∈ C 2(O) : (ϕ− v)(x) = max

O(ϕ− v)

Av(x) :=

ϕ ∈ C 2(O) : (ϕ− v)(x) = min

O(ϕ− v)

Definition (i) v ∈ LSC(O) is a viscosity supersolution (resp.subsolution) of (E) if

−g(x , v(x),Dϕ(x),D2ϕ(x)

)≥ (resp. ≤ ) 0

(ii) v ∈ C 0(O) is a viscosity solution if both viscosity super solutionand subsolution

Uniqueness implied by comparison result (maximum principle) :

v1 subsol, v2 supersol, with v1 ≤ v2 on ∂O −→ v1 ≤ v2 on cl(O)





Back to the heat equation

Tower property =⇒ v(t, x) := EP0[g(x + Bt

T )]satisfies

u(t, x) = EP0[u(τ, x+Bt

τ )]for all stopping time τ ∈ [t,T ], P0−a.s.

Supersolution For (t, x) and ϕ ∈ Av(t, x), i.e.

w.l.o.g. 0 = (ϕ− v)(t, x) = max(ϕ− v)

we compute that

ϕ(t, x) = u(t, x) = EP0[u(τ, x + Bt

τ )]≥ EP0

[ϕ(τ, x + Bt

τ )]

Itô’s formula applies to ϕ, send τ t =⇒

−∂tϕ(t, x)− 12D2ϕ(t, x) ≥ 0

Subsolution : same argumentNizar TOUZI Viscosity Solutions of PPDEs






T )]satisfies




w.l.o.g. 0 = (ϕ− v)(t, x) = max(ϕ− v)

we compute that

ϕ(t, x) = u(t, x) = EP0[u(τ, x + Bt

τ )]≥ EP0

[ϕ(τ, x + Bt

τ )]


−∂tϕ(t, x)− 12D2ϕ(t, x) ≥ 0







T )]satisfies




w.l.o.g. 0 = (ϕ− v)(t, x) = max(ϕ− v)

we compute that

ϕ(t, x) = u(t, x) = EP0[u(τ, x + Bt

τ )]≥ EP0

[ϕ(τ, x + Bt

τ )]


−∂tϕ(t, x)− 12D2ϕ(t, x) ≥ 0





Our objective

Develop a notion of viscosity solution of nonlinearpath-dependent PDEs

Major difficulty : standard theory of viscosity solutions is based onlocal compactness of the underlying space...

However Ω = ω ∈ C 0([0,T ],Rd) : ω(0) = 0 not locally compact

We introduce a new definitionWe provide some first existence and uniqueness results



Elliptic path-dependent PDEsConsistency, stability, partial comparison

Outline







Smooth test processes

Nonlinear expectation ELt := supP∈PLtEP and ELt := infP∈PL

tEP

Nonlinear Snell envelope SLt [X ](ω) := supτ∈T t ELt[X t,ωτ

]and SLt · · ·

Smooth test processes

ALu(t, ω) :=ϕ ∈ C 1,2(Λt) : (ϕ−ut,ω)t(0) = SLt

[(ϕ−ut,ω).∧h

]for some h ∈ Ht

ALu(t, ω) :=

ϕ ∈ C 1,2(Λt) : (ϕ−ut,ω)t(0) = SLt

[(ϕ−ut,ω).∧h

]for some h ∈ Ht




Definition

u ∈ U (resp. U) is a viscosity...• L-subsolution (resp. L-supersolution) of PPDE if :

−∂tϕt(0)− G(t, ω, u(t, ω), ∂ωϕt(0), ∂2

ωωϕt(0))≤ (resp. ≥) 0

for all (t, ω) ∈ [0,T )× Ω and ϕ ∈ ALu(t, ω) (resp. ϕ ∈ ALu(t, ω))

• subsolution (resp. supersolution) of PPDE if ∃ L > 0 s.t. u isviscosity L-subsolution (resp. L-supersolution) of PPDE

• solution of PPDE if it is both a viscosity subsolution and aviscosity supersolution




Definition


−∂tϕt(0)− G(t, ω, u(t, ω), ∂ωϕt(0), ∂2

ωωϕt(0))≤ (resp. ≥) 0







Definition


−∂tϕt(0)− G(t, ω, u(t, ω), ∂ωϕt(0), ∂2

ωωϕt(0))≤ (resp. ≥) 0







Consistency with classical solutions

Assumption G1 G (t, ω, y , z , γ) nondecreasing in γ and satisfies :(i) G (·, y , z , γ) is F-prog. meas., and ‖G (·, 0, 0, 0)‖∞ <∞.(ii) G is uniformly continuous in ω(iii) G is uniformly Lipschitz in (y , z , γ)

Theorem

Let Assumption G1 hold and u ∈ C 1,2b (Λ). Then the following

assertions are equivalent :u classical solution (resp. subsolution, supersolution) of PPDEu viscosity solution (resp. subsolution, supersolution) of PPDE




Stability

Consider the perturbed PPDEs

PPDEε : G ε(., u, ∂ωu, ∂

2ωωu

)= 0 on Λ

TheoremLet uε viscosity L−subsolution (resp. L−supersolution) ofPPDE(G ε), for some fixed L > 0. Assume

(G ε, uε) −→ (G , u) as ε→ 0, loc. unif. in Λ.

Then u is a viscosity L−subsolution (resp. supersolution) of PPDEwith coefficient G .




Partial comparison

TheoremLet Assumption G1 hold. Let

u1 be a bounded viscosity subsolution of PPDE,u2 a bounded viscosity supersolution of PPDE,u1(T , ·) ≤ u2(T , ·).

Assume further that either u1 or u2 is in C 1,2(Λ). Then

u1 ≤ u2 on Λ

Proofs of all previous results are easy application of optimalstopping theory : Snell envelop is a supermartingale, and the firsthitting time of the obstacle is an optimal stopping rule




Partial comparison

TheoremLet Assumption G1 hold. Let

u1 be a bounded viscosity subsolution of PPDE,u2 a bounded viscosity supersolution of PPDE,u1(T , ·) ≤ u2(T , ·).

Assume further that either u1 or u2 is in C 1,2(Λ). Then

u1 ≤ u2 on Λ

Proofs of all previous results are easy application of optimalstopping theory : Snell envelop is a supermartingale, and the firsthitting time of the obstacle is an optimal stopping rule




Additional assumptions

Assumption G2 Either one of the following conditions :(i) G convex in γ and uniformly elliptic,(ii) or, G is convex in (y , z , γ)(iii) or, d ≤ 2

Allows to apply standard PDE theory to a conveniently definedpath-frozen equation...




Freezing ω in the generator

• Define the deterministic function on [t,T ]× R× Rd × Sd :

g t,ω(s, y , z , γ) := G (s, ω·∧t , y , z , γ)

• Consider the standard PDE :

Lt,ωv := −∂tv − g t,ω(s, v ,Dv ,D2v) = 0, (t, x) ∈ Oε,ηt

where

Oε,ηt := [t, (1 + η)T )× x ∈ Rd : |x | < ε, ε > 0, η ≥ 0




The main results

Theorem (Comparison)

Under Assumptions G1-G2, let u1 ∈ U , u2 ∈ U , ξ ∈ UCb(Ω) s.tu1 is a bounded viscosity subsolution of PPDEu2 is a bounded viscosity supersolution of PPDEu1(T , ·) ≤ ξ ≤ u2(T , ·)

Then u1 ≤ u2 on Λ.

Theorem (Existence)

Under Assumptions G1, G2, for any ξ ∈ UCb(Ω), the PPDE withterminal condition ξ has a unique bounded viscosity solutionu ∈ UCb(Λ).




The main results

Theorem (Comparison)

Under Assumptions G1-G2, let u1 ∈ U , u2 ∈ U , ξ ∈ UCb(Ω) s.tu1 is a bounded viscosity subsolution of PPDEu2 is a bounded viscosity supersolution of PPDEu1(T , ·) ≤ ξ ≤ u2(T , ·)

Then u1 ≤ u2 on Λ.

Theorem (Existence)

Under Assumptions G1, G2, for any ξ ∈ UCb(Ω), the PPDE withterminal condition ξ has a unique bounded viscosity solutionu ∈ UCb(Λ).




FormulationA purely probabilistic comparison result

Outline








Our objective : nonlinear path-dependent PDEs

Let O bounded open subset of Rd , and

Ωe :=ω.∧t ∈ Ω : t ≥ 0 and ω.∧t ∈ O

Find prog. meas. process

u(t, ω) = u(ω.∧t), i.e. no t−dependence

satisfying the elliptic equation :

−G (., u, ∂ωu, ∂2ωωu)

(ω.∧t) = 0, ω ∈ Ωe ,

u = ξ on ∂Ωe





A distance accounting for ellipticity

Z.-J. Ren introduced

de(ω, ω′) := infπsupt

∣∣ωt − ωπ(t)∣∣,

π : one-to-one increasing maps

• For a good formulation of the problem G , ξ and u must becontinuous in the sense of de ...

• Previous results stated for parabolic PDEs extend to the ellipticcase (Ren ’13)





Main result

Consider the elliptic path-dependent PDE :

(E − PPDE ) − G(∂2ωωu

)= 0, ω ∈ Ωe

G : Sd −→ R Lipschitz and uniformly elliptic

TheoremLet u, v ∈ UCb(Ωe) be viscosity subsolution and super solution,respectively, of (E-PPDE). Then

u ≤ v on ∂Ωe =⇒ u ≤ v on cl(Ωe)





REST OF THE TALK

• Sketch the proof in the case G = Trace

• The proof is an adaptation to the probabilistic framework ofarguments of Luis Cafarelli and Xavier Cabre

• We believe that the present probabilistic proof is easier. Indeedour class of test processes is larger !

Existence may get harder, but our definition is essentiallysharp...More restrictions give better chances for uniqueness...





Punctual Laplacian

For ϕ ∈ UCb(Ωe), define

J +ϕ(ωt∧.) :=a : u(ωt∧.) = maxτ≥0 E

[u(ω ⊗t Bτ∧h∧.)− 1

2a(τ ∧ h)]

J −ϕ(ωt∧.) :=a : u(ωt∧.) = minτ≥0 E

[u(ω ⊗t Bτ∧h∧.)− 1

2a(τ ∧ h)]

• For a1 ∈ J +ϕ(ωt∧.) and a2 ∈ J −u(ωt∧.), we have a1 ≥ a2

Definition ϕ is punctually C 2 at ωt∧. if

cl[J+ϕ(ωt∧.)

]∩ cl[J−ϕ(ωt∧.)

]= a 6= ∅

We denote ∆ωϕ(ωt∧) := a





Punctual Laplacian and viscosity solution

Proposition Let u be a viscosity subsolution of the path-dependentLaplace equation. For ωt∧. ∈ Ωe , we have :

u punctually C 2 at ωt∧. =⇒ −∆ωu(ωt∧.) ≤ 0

A similar statement holds for supersolutions.





An adaptation of the Jensen regularization

• In standard viscosity solutions, the Jensen regularizations

vn(x) := infyv(y) + n|x − y |2 and un(x) := sup

yu(y)− n|x − y |2

play a crucial role (semi-convex/concave, Aleksandrov, etc...)

• In our path-dependent case, we introduce

vn(ωt∧.) := infτE[v(Bt,ω

τ∧HO) + n|Bt,ω

τ∧HO− ωt |2

]un(ωt∧.) := sup

τE[u(Bt,ω

τ∧HO)− n|Bt,ω

τ∧HO− ωt |2

]where HO := infs > t : Bt,ω

s 6∈ O.





Properties of the regularization

• un −→ u and vn −→ v pointwise

• By optimal stopping theory, vn(ωt∧.) + nt submartingale, so

vn(ωt∧.) + nt = vn0 + At + Mt , A, M martingale

• vn is a viscosity solution of −∂2ωωv

n ≥ 0

THEN : At =∫ t0 asds, vn is punctually C 2, and

at =12

∆ωvn(ωt∧.) + nω2t

Similar statements for un holds





Proof of the comparison result

• un and vn are viscosity supersolution and subsolution,respectively, of the Laplace equation

and finally,

• un − vn is a subsolution of the Laplace equation :

−Tr[∂2ωω(un − vn)

]≤ 0

• Since 0 is a classical supersolution of the Laplace equation, wededuce from the partial comparison result that un − vn ≤ 0, andsending n→∞ :

u ≤ v





More progress is needed :

• General elliptic equations

• Parabolic equation

THANK YOU FOR YOUR ATTENTION


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Viscosity Solutions of Fully Nonlinear Path Dependent PDEs