On viscosity solutions of path dependent PDEs

Ibrahim Ekren · Christian Keller · Nizar Touzi · Jianfeng Zhang

181 citations0/0 files storedThe Annals of Probability

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Fuzzy Systems and Optimization

Abstract

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional Itô calculus recently introduced by Dupire [Functional Itô calculus (2009) Preprint].

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References

2005Controlled Markov Processes and Viscosity Solutions 1990Adapted solution of a backward stochastic differential equation 1997Backward Stochastic Differential Equations in Finance 2005Backward stochastic differential equations and quasilinear parabolic partial differential equations 1994Solving forward-backward stochastic differential equations explicitly — a four step scheme 1997Reflected solutions of backward SDE's, and related obstacle problems for PDE's 1999Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type 1999Forward-backward stochastic differential equations and quasilinear parabolic PDEs

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