Well-posedness of monotone semilinear SPDEs with semimartingale noise

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, accommodating highly general monotone nonlinearities and path-dependent multiplicative noise.

Contribution

It extends the theory of SPDEs by allowing very general monotone nonlinearities and path-dependent multiplicative noise, using a novel approach with a priori estimates and an infinite-dimensional Doob inequality.

Findings

Proved existence and uniqueness of solutions under mild conditions.

Handled highly general monotone nonlinearities with arbitrary growth.

Developed a new method using a Doob-type inequality for infinite dimensions.

Abstract

We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to M\'etivier and Pellaumail.