Representation and approximation of ambit fields in Hilbert space

TL;DR

This paper introduces Hambit fields, a new class of Hilbert space-valued stochastic processes, and develops their mathematical properties, including a PDE interpretation and a convergent numerical scheme.

Contribution

It extends ambit fields to Hilbert spaces, providing a new framework with PDE interpretation and a finite difference scheme for approximation.

Findings

Hambit fields can be expressed as sums of weighted real-valued processes.

They can be viewed as boundaries of solutions to certain SPDEs.

A convergent finite difference scheme for Hambit fields is developed.

Abstract

We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state space of the Hambit field. We provide an explicit construction of such a space. Finally, we apply this interpretation of Hambit fields to develop a finite difference scheme, for which we prove convergence under some Lipschitz conditions.