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 $\min(3α-1,1)$, 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.

• The paper presents a kernel-integrated Euler scheme that rigorously quantifies weak error rates for rough and mean-reverting Gaussian volatility models.
• It employs fractional and Malliavin calculus to derive explicit convergence rates using fractional convolution kernels and special functions.
• The analysis offers practical implications for simulation-based finance, ensuring uniform, time-consistent error bounds for advanced volatility approximations.

Weak Error Approximation for Rough and Gaussian Mean-Reverting Stochastic Volatility Models

Model Formulation and Analytical Extensions

The paper investigates weak error rates for Euler-type discretization schemes applied to rough stochastic volatility models that generalize the Stein-Stein framework by incorporating mean-reverting Gaussian volatility driven by stochastic Volterra equations (SVEs). The volatility dynamics are parameterized using fractional convolution kernels $\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}$ with $\alpha \in (1/2,1)$, embedding non-Markovian and long-memory effects in the structure, and allow for stationary limiting distributions when the mean-reversion parameter $\kappa_2 < 0$.

The SVE for the volatility process $X_t$ is structured as:

$$X_t = x_0 + \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} (\kappa_1 + \kappa_2 X_s) ds + \sigma \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} dW_s,$$

with the asset log-price incorporating both drift and volatility feedback. This structure is broad enough to contain both classical and rough Stein-Stein models, allowing for stationary volatility when $\kappa_2<0$. The authors also provide explicit formulas for expectations and covariances in terms of Mittag-Leffler functions, proving the existence of Gaussian stationary distributions and establishing the Malliavin differentiability of the involved processes.

Weak Error Analysis for Euler Schemes

The main contribution is a rigorous quantification of weak error rates for discretized Euler schemes applied to these models, considering both volatility discretization and asset price approximation. The kernel-integrated Euler scheme is analyzed in-depth, distinguishing it from other naive discretization approaches that yield suboptimal rates. The authors precisely characterize weak convergence rates for smooth polynomial test functions $\Phi$, demonstrating that the error is governed by:

$${\bf v}_n(\alpha) = \begin{cases} 1/n & \text{if } \alpha \in (2/3,1) \ \log(n)/n & \text{if } \alpha = 2/3 \ 1/n^{3\alpha-1} & \text{if } \alpha \in (1/2,2/3) \end{cases}$$

This generalizes and unifies prior results for rough volatility models by rigorously extending them to cases with non-trivial mean-reversion and drift parameters.

Strong functional analytic and probabilistic tools are leveraged, including: fractional calculus, Malliavin calculus (Clark-Ocone representation for weak errors), and detailed combinatorial expansions (word-based moment formulas, extending [FSW, Gassiat]). Technical estimates on convolutions and special functions (Beta, Gamma, hypergeometric) are systematically employed to bound approximation terms, and Gronwall-type arguments are applied for recursive error propagation.

Numerical Results and Implications

For polynomial test functions, the weak error for the discretized rough OU volatility process is shown to converge at order ${\bf v}_n(\alpha)$, robustly extending prior results to settings including non-zero mean reversion and drift terms ($\kappa_1, \kappa_2, b$ arbitrary). The authors establish that when $\kappa_2\neq 0$, discretization introduces genuine volatility approximation error, but their analysis proves that the convergence rates persist and are unaffected in the weak sense.

This result implies that even in non-Markovian, mean-reverting rough stochastic volatility environments (with parameters relevant to real financial models), robust numerical approximations via kernel-integrated Euler schemes are available, with quantifiable error rates. The convergence bounds are explicit and uniform in time, and the stationary distribution and its moments are tractable. These findings strongly motivate the deployment of such discretization techniques in large-scale simulation of rough volatility models, including scenario generation, Monte Carlo parametric sensitivity, and multi-asset extensions.

Theoretical Impact and Future Directions

The paper advances a comprehensive analytic framework for weak convergence of numerical schemes in rough Gaussian volatility SVEs, confirming the persistence of sharp rates up to full Stein-Stein generality. By extending the recursive martingale/polyomial moment expansions (via Malliavin calculus and word combinatorics), the results bridge strong and weak error regimes and provide a toolkit for further investigations in path-dependent PDEs and high-dimensional settings.

The framework allows for stationary distributions (by negative mean-reversion) and is well-suited for sensitivity analysis (including multi-parameter joint sampling), mitigating pitfalls in naive covariance calculation. The technical apparatus is generic and can be adapted for broader classes of SVEs, including those arising in rough path theory and multifractal modeling.

Prospective avenues include:

• Application to option pricing under rough volatility with general mean-reversion
• Extension to multidimensional SVEs and non-Gaussian increments
• Weak error control for adaptive and higher-order schemes in rough volatility
• Detailed study of pathwise convergence for functionals beyond polynomials.

Conclusion

This work establishes rigorous quantitative weak error rates for Euler-type discretizations of rough Gaussian mean-reverting stochastic volatility models, encompassing general Stein-Stein variants, and providing explicit analytic bounds for practical polynomial functionals. The convergence rates are shown to persist under general parameterizations and discretization errors, with direct applications in simulation-based finance and stochastic modeling, and open theoretical directions for further elaboration of numerical methods for SVEs in rough volatility settings (2602.18234).