Markovian Lifts of Stochastic Volterra Equations in Sobolev Spaces: Solution theory, an Ito Formula and Invariant Measures

2024/06/14 by Florian Huber, Huber, Florian
Economics, Econometrics and Finance · #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2406.10352

Abstract

We investigate Markovian lifts of stochastic Volterra equations (SVEs) with completely monotone kernels and general coefficients within the framework of weighted Sobolev spaces. Our primary focus is developing a comprehensive solution theory for a class of non-local stochastic evolution equations (SEEs) encompassing these Markovian lifts. This enables us to provide conditions for the existence of invariant measures for the lifted processes and the corresponding SVE, and, for uniformly elliptic diffusion coefficients, to establish uniqueness and exponential ergodicity via a generalized Harris theorem. Another key contribution is an It^o-type formula for the stochastic Volterra equations under consideration, from which we develop a range of applications: a finite-time blow-up criterion, a Feynman-Kac representation together with the associated backward Kolmogorov equation on the lift space, and a pricing equation for rough volatility models.

Related

• Limit theorems for stochastic Volterra processes
• Kolmogorov equations for stochastic Volterra processes with singular kernels
• Comparison principles for stochastic Volterra equations
• Markovian lifting and optimal control for integral stochastic Volterra equations with completely monotone kernels
• Failure of the Markov property for stochastic Volterra equations