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

Aurélien Alfonsi, Ahmed Kebaier

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 $\min(3α-1,1)$, where $\frac{t^{α-1}}{Γ(α)} $ is the fractional convolution kernel with $α\in (1/2,1)$. Then, our main result is to obtain the same convergence rate for the corresponding stochastic rough volatility model with polynomial test functions.

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

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 , where is the fractional convolution kernel with . Then, our main result is to obtain the same convergence rate for the corresponding stochastic rough volatility model with polynomial test functions.
Paper Structure (13 sections, 23 theorems, 244 equations)

This paper contains 13 sections, 23 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Convergence for the rough Volterra Ornstein-Uhlenbeck process
  3. The rough Ornstein-Uhlenbeck process
  4. Weak rate of convergence
  5. Weak error approximation with the cubic test function
  6. Weak error approximation with polynomial test functions
  7. Representation of the weak error
  8. Weak error analysis
  9. Analysis of $\mathcal{E}_{1,1}$ and $\mathcal{E}_{1,2}$
  10. Analysis of $\mathcal{E}_{2,1}$
  11. Analysis of $\mathcal{E}_{2,2}$
  12. Technical results
  13. Technical proofs

Key Result

Proposition 2.1

Let $X$ be the solution of the Stochastic Volterra Equation SVE_OU. Then, we have where $E_\alpha(z)=\sum_{i=0}^\infty \frac{z^i}{\Gamma(\alpha i +1)}$ is the Mittag-Leffler function.

Theorems & Definitions (51)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • Theorem 2.7
  • ...and 41 more