Weak error approximation for rough and Gaussian mean-reverting stochastic volatility models

TL;DR

This paper investigates the weak approximation rate of Euler schemes for rough and Gaussian mean-reverting stochastic volatility models, including the rough Stein-Stein model, establishing convergence rates dependent on the roughness parameter.

Contribution

It provides the first weak convergence rate results for discretized rough Ornstein-Uhlenbeck processes and extends these results to rough volatility models with polynomial test functions.

Findings

Weak convergence rate for rough Ornstein-Uhlenbeck process is min(3α-1,1).

Same convergence rate applies to rough volatility models with polynomial test functions.

Results depend on the fractional convolution kernel parameter α.

Abstract

For a class of stochastic models with Gaussian and rough mean-reverting volatility that embeds the genuine rough Stein-Stein model, we study the weak approximation rate when using a Euler type scheme with integrated kernels. Our first result is a weak convergence rate for the discretised rough Ornstein-Uhlenbeck process, that is essentially in $\min(3\alpha-1,1)$, where $\frac{t^{\alpha-1}}{\Gamma(\alpha)} $ is the fractional convolution kernel with $\alpha \in (1/2,1)$. Then, our main result is to obtain the same convergence rate for the corresponding stochastic rough volatility model with polynomial test functions.