Convergence in Multiscale Financial Models with Non-Gaussian Stochastic Volatility

TL;DR

This paper investigates the asymptotic behavior of multiscale financial models with non-Gaussian stochastic volatility driven by jump processes, using PDE methods to analyze the convergence as the time scale separation parameter approaches zero.

Contribution

It introduces a new analysis of singular perturbations in financial models with jump-driven volatility, extending classical results to non-Gaussian settings.

Findings

Established convergence of the stochastic control systems as the scale parameter tends to zero.

Developed viscosity solution techniques for PDEs associated with jump-driven volatility models.

Provided insights into the impact of non-Gaussian jumps on financial model asymptotics.

Abstract

We consider stochastic control systems affected by a fast mean reverting volatility $Y(t)$ driven by a pure jump L\'evy process. Motivated by a large literature on financial models, we assume that $Y(t)$ evolves at a faster time scale $\frac{t}{\varepsilon}$ than the assets, and we study the asymptotics as $\varepsilon\to 0$. This is a singular perturbation problem that we study mostly by PDE methods within the theory of viscosity solutions.