On the uniqueness of solutions of stochastic Volterra equations

TL;DR

This paper establishes strong existence, uniqueness, and regularity results for a broad class of stochastic Volterra equations with singular kernels and non-Lipschitz diffusion, extending classical theorems and applying to financial models.

Contribution

It extends Yamada-Watanabe's theorem to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, providing new theoretical foundations.

Findings

Proves strong existence and uniqueness for the class of equations.

Establishes Hölder regularity of solutions.

Applies results to the rough Heston model in finance.

Abstract

We prove strong existence and uniqueness, and H\"older regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised kernels. We apply these results to the rough Heston model, with square-root diffusion coefficient, recently proposed in Mathematical Finance to model the volatility of asset prices.