Numerical convergence for the singular stochastic integral equation driving rough volatility

Develop and prove convergence results for numerical methods that approximate the singular stochastic integral equation Y(t) = (\bar{ν}/Γ(α)) ∫_0^t (t−s)^{α−1}(v_1^⊤ 𝟙 − Y(s)) ds + (\bar{κ}\bar{ν}/Γ(α)) ∫_0^t (t−s)^{α−1} √Y(s) dB_1(s) (equation (eq20241113_2)), despite the failure of standard Lipschitz and linear-growth assumptions used in existing analyses.

Background

The rough-volatility factor Y(t) satisfies a fractional-kernel stochastic Volterra equation with square-root diffusion. The authors review minimal assumptions from the literature under which numerical convergence can be proved and explain that their equation does not satisfy these assumptions.

Consequently, the convergence theory for numerical schemes applied to this SSIE is not covered by existing results and is posed as an explicit open problem.