Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations

Abstract

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

Keywords:

• Stochastic partial differential equation
• limiting law
• Hilbert space
• Lévy noise
• stochastic Volterra integral equation

Acknowledgement

The authors are grateful to Bernt Øksendal for proposing this topic.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 If $\mathcal{R}$ is the RKHS of L, cf. [20, Definition 7.2], then $\mathcal{R}\subseteq \mathcal{V}$ and for $T\in L(\mathcal{V},\mathcal{U})$ one has $\Vert T{\Vert }_{L_2(\mathcal{R},\mathcal{U})}\le \Vert T{\Vert }_{\mathrm{o}\mathrm{p}}$.

Additional information

Funding

F. E. Benth acknowledges financial support from 'FINEWSTOCH', funded by the Norwegian Research Council.