Markovian Integral Equations

Abstract

We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman-Kac formula and a one-dimensional global existence- and uniqueness result.

References in corpus (4)

• On viscosity solutions of path dependent PDEs
• A control problem with fuel constraint and Dawson-Watanabe superprocesses
• On the support of solutions of stochastic differential equations with path-dependent coefficients
• Mild and viscosity solutions to semilinear parabolic path-dependent PDEs

Cited by in corpus (2)

• On existence and uniqueness properties for solutions of stochastic fixed point equations
• Mild to classical solutions for XVA equations under stochastic volatility