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Viscosity Solutions of Path-Dependent PDEs with Randomized Time

18 citations0/0 files storedSIAM Journal on Mathematical Analysis

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Differential Equations and Boundary Problems

Abstract

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204--236]. With the new definition, we prove the two important results, until now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/supersolutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.

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References

2013Functional Itô calculus and stochastic integral representation of martingales 2014On viscosity solutions of path dependent PDEs 2016Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II 2013Constructing sublinear expectations on path space 2016Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I 2014Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman--Isaacs Equation 2015BSDE, path-dependent PDE and nonlinear Feynman-Kac formula 2014Optimal stopping under nonlinear expectation

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