Cubature Method for Stochastic Volterra Integral Equations

TL;DR

This paper develops a cubature method for solving Stochastic Volterra Integral Equations, offering a more efficient alternative to Euler schemes through explicit measure construction and rigorous error estimates.

Contribution

It introduces a novel cubature formula for stochastic Volterra equations, including derivation, explicit measure construction, and efficiency analysis.

Findings

Cubature method outperforms Euler scheme in efficiency.

Explicit construction of cubature measure for special cases.

Rigorous error estimates provided for the method.

Abstract

In this paper, we introduce the cubature formula for Stochastic Volterra Integral Equations. We first derive the stochastic Taylor expansion in this setting, by utilizing a functional It\^{o} formula, and provide its tail estimates. We then introduce the cubature measure for such equations, and construct it explicitly in some special cases, including a long memory stochastic volatility model. We shall provide the error estimate rigorously. Our numerical examples show that the cubature method is much more efficient than the Euler scheme, provided certain conditions are satisfied.