Viscosity Solutions of Fully Nonlinear Path Dependent PDEs Nizar - PowerPoint PPT Presentation

Motivation and examples Preliminaries Fully nonlinear path dependent PDE Wellposedness Viscosity Solutions of Fully Nonlinear Path Dependent PDEs Nizar TOUZI Ecole Polytechnique France Joint work with Ibrahim EKREN and Jianfeng ZHANG Happy Birthday Ioannis Columbia University, June 4, 2012 Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Outline 1 Motivation and examples Parabolic nonlinear PPDEs Examples Intuition 2 Preliminaries 3 Fully nonlinear path dependent PDE PPDE in Ω Definition of viscosity solutions First properties 4 Wellposedness Additional assumption Existence and uniqueness Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Parabolic nonlinear path-dependent PDEs Let Ω = { ω ∈ C 0 ([ 0 , T ] , R d ) , ω 0 = 0 } , B canonical process, Φ the corresponding filtration Our objective : wellposedness theory for the equation : � � − ∂ t u − F ( ., u , ∂ ω u , ∂ ωω u ) ( t , ω ) = 0 for t < T , ω ∈ Ω u ( T , ω ) = g ( ω ) � � where g ( ω ) = g ( ω s ) s ≤ T and F ( t , ω, y , z , γ ) is F − prog. meas. map : F : [ 0 , T ] × Ω × R × R d × S d − → R and the unknown process u ( t , ω ) is prog. meas. Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Time and space derivatives • Time derivative introduced by Dupire : u ( t + h , ω t ∧ . ) − u ( t , ω ) ∂ t u ( t , ω ) := lim if exists h h ց 0 • Space derivatives : u ( t , ω ) ∈ C 1 , 2 if there exist continuous process, denoted ∂ ω u , ∂ ωω u , such that Itô’s formula holds : ∂ t udt + 1 du = 2 ∂ ωω ud � B � + ∂ ω dB ( ... ) Remark If ∂ t u , ∂ ω u , ∂ ωω u in Dupire sense exist and continuous bounded, then Itô’s formula holds true Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Example : Backward SDE Find F − progressively measurable processes ( Y , Z ) such that : � T � T Y t = g − f ( s , ω, Y s , Z s ) ds + Z s dB s t t � T 0 | f t ( 0 , 0 ) | 2 dt < ∞ , the Pardoux and Peng 1990 : f Lipschitz, E is a unique solution ( Y , Z ) in the space � T | Y t | 2 < ∞ | Z t | 2 dt < ∞ � Y � S 2 := E sup and � Y � H 2 := E t ≤ T t If Y ( t , ω ) ∈ C 1 , 2 , then Z t = ∂ ω Y t and ∂ t Y + 1 2 ∂ ωω Y = − f ( ., Y , ∂ ω Y ) and Y ( T , . ) = g Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Example 2 : Second order BSDE • P ∈ P := { P σ : σ ≤ σ ≤ σ } : dW σ t := σ − 1 t dB t is a P σ -Brownian motion and W σ and B induce the same P σ -augmented filtration. • Second order BSDE (Cheridito-Soner-T.-Victoir 07, Soner-T.-Zhang 11) : dY t = − f ( t , ω, Y t , Z t , σ 2 t ) dt + Z t dB t − dK t , Y T = g ( ω ) , P σ -a.s. for all σ. ⋄ When f = 0, Y is a G -martingale (Peng 06) • If Y ( t , ω ) ∈ C 1 , 2 , then Z t = ∂ ω Y t , and ∂ t Y + F ( t , ω, Y , ∂ ω Y , ∂ ωω Y ) = 0 , Y ( T , . ) = g � � 1 2 σ 2 γ + f ( t , ω, y , z , σ 2 ) F ( t , ω, y , z , γ ) := sup σ ≤ σ ≤ σ . Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness A larger class of fully nonlinear PPDEs • Dynamic programming equation for a non-Markov stochastic control problem • Dynamic programming equations for non-Markov differential games Not accessible from the existing 2BSDE results Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Example 3 : Backward Stochastic PDE • BSPDE : du ( t , x , ω ) = − f ( t , x , ω, u , ∂ x u , ∂ xx u , β, ∂ x β ) dt + β ( t , x , ω ) dB t • Functional Itô’s formula : du ( t , x , ω ) = ( ∂ t u + 1 2 ∂ ωω u ) dt + ∂ ω udB t Then β = ∂ ω u , β x = ∂ x ω u , and we arrive at the PPDE : ∂ t u + 1 2 ∂ ωω u + f ( t , x , ω, u , ∂ x u , ∂ xx u , ∂ ω u , ∂ x ω u ) = 0 . Our approach allows to handle General mixed PPDE : ∂ t u + F ( t , x , ω, u , ∂ x u , ∂ ω u , ∂ xx u , ∂ x ω u , ∂ ωω u ) = 0 u ( T , x , ω ) = g ( x , ω ) . Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Applications of BSPDEs • Solving non-Markov FBSDEs by the method of decoupling field (Zhang 06, Ma-Yin-Zhang 10, Ma-Wu-Zhang-Zhang 10) • Control of Stochastic PDEs • Rate function for a large deviation problem (Ma-T.-Zhang) ∂ t u − 1 � 2 = 0 , u ( T , x , ω ) = g ( x , ω ) � � � ∂ ω u + ∂ x u σ ( t , x , ω ) 2 (Path-dependent Eikonal equation) Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Why viscosity solutions of PPDEs We want to adapt the theory of viscosity solutions to the present case • To obtain wellposedness for a larger class of equations • Powerful stability result • Easy ! Main difficulty : the paths space Ω is not locally compact Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Recall standard viscosity solutions f ( x , y , z , γ ) nondecreasing in γ . Consider the PDE : � � − f ( ., v , Dv , D 2 v ) (open subset of R d ) ( E ) ( x ) = 0 , x ∈ O For v ∈ C 2 ( O ) , the following are equivalent : Exercise (i) v is a supersolution of (E) (ii) For all ( x 0 , φ ) ∈ O × C 2 ( O ) : � � − f ( ., v , D φ, D 2 φ ) ( φ − v )( x 0 ) = max O ( φ − v ) = ⇒ ( x 0 ) ≥ 0 Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Intuition from consistency with classical solutions (1) Since v ( t , x ) is a classical supersolution : � � − ∂ t v − F ( ., v , Dv , D 2 v ) 0 ≤ ( t 0 , x 0 ) � � − ∂ t φ − F ( ., v , D φ, D 2 φ ) = ( t 0 , x 0 ) � � + F γ ( ... ) D 2 ( φ − v ) + ∂ t ( φ − v ) + F z ( ... ) D ( φ − v ) ( t 0 , x 0 ) � �� � � �� � � �� � = 0 = 0 ≤ 0 � � − ∂ t φ − F ( ., v , D φ, D 2 φ ) ≤ ( t 0 , x 0 ) by F γ ≥ 0, the first and second order conditions for ( φ − v )( x 0 ) = max O ( φ − v ) Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Intuition from consistency with classical solutions (2) Since u ( t , ω ) is a classical supersolution : � � 0 ≤ − ∂ t u − F ( ., u , ∂ ω u , ∂ ωω u ) ( t 0 , ω 0 ) � � = − ∂ t ϕ − F ( ., u , ∂ ω ϕ, ∂ ωω ϕ ) ( t 0 , ω 0 ) � � + ∂ t ( ϕ − u ) + F z ( ... ) ∂ ω ( ϕ − u ) + F γ ( ... ) ∂ ωω ( ϕ − u ) ( t 0 , ω 0 ) � �� � =: R ( t 0 ,ω 0 ) ≤ 0 Needed � � ≤ − ∂ t ϕ − F ( ., u , ∂ ω ϕ, ∂ ωω ϕ ) ( t 0 , ω 0 ) Remark d ( ϕ − u )( t 0 , ω 0 ) = R ( t 0 , ω 0 ) dt + ∂ ω ( ϕ − u )( t 0 , ω 0 ) dB , ˆ P − a.s. where ˆ P is the probability measure on Ω under which � α t dt + ˆ β t d ˆ ˆ dB t = ˆ W t , α := F p ( ... ) , ˆ β := 2 F γ ( ... ) W is a ˆ ˆ P − Brownian motion Nizar TOUZI Viscosity Solutions of PPDEs

Motivation and examples Parabolic nonlinear PPDEs Preliminaries Examples Fully nonlinear path dependent PDE Intuition Wellposedness Intuition from consistency with classical solutions (2) Since u ( t , ω ) is a classical supersolution : � � 0 ≤ − ∂ t u − F ( ., u , ∂ ω u , ∂ ωω u ) ( t 0 , ω 0 ) � � = − ∂ t ϕ − F ( ., u , ∂ ω ϕ, ∂ ωω ϕ ) ( t 0 , ω 0 ) � � + ∂ t ( ϕ − u ) + F z ( ... ) ∂ ω ( ϕ − u ) + F γ ( ... ) ∂ ωω ( ϕ − u ) ( t 0 , ω 0 ) � �� � =: R ( t 0 ,ω 0 ) ≤ 0 Needed � � ≤ − ∂ t ϕ − F ( ., u , ∂ ω ϕ, ∂ ωω ϕ ) ( t 0 , ω 0 ) Remark d ( ϕ − u )( t 0 , ω 0 ) = R ( t 0 , ω 0 ) dt + ∂ ω ( ϕ − u )( t 0 , ω 0 ) dB , ˆ P − a.s. where ˆ P is the probability measure on Ω under which � α t dt + ˆ β t d ˆ ˆ dB t = ˆ W t , α := F p ( ... ) , ˆ β := 2 F γ ( ... ) W is a ˆ ˆ P − Brownian motion Nizar TOUZI Viscosity Solutions of PPDEs