The Averaging Principle for Non-autonomous Slow-fast Stochastic Differential Equations and an Application to a Local Stochastic Volatility Model

TL;DR

This paper establishes an averaging principle for non-autonomous slow-fast stochastic differential equations, proving weak convergence to an averaged system, and applies it to a local stochastic volatility model in finance.

Contribution

It extends the averaging principle to non-autonomous stochastic systems with specific regularity conditions and demonstrates its application in financial modeling of stochastic volatility.

Findings

Proved weak convergence of slow component to averaged equation.

Applied the theory to a local stochastic volatility model.

Demonstrated convergence of derivative prices under the model.

Abstract

In this work we study the averaging principle for non-autonomous slow-fast systems of stochastic differential equations. In particular in the first part we prove the averaging principle assuming the sublinearity, the Lipschitzianity and the Holder's continuity in time of the coefficients, an ergodic hypothesis and an $\mathcal{L}^2$-bound of the fast component. In this setting we prove the weak convergence of the slow component to the solution of the averaged equation. Moreover we provide a suitable dissipativity condition under which the ergodic hypothesis and the $\mathcal{L}^2$-bound of the fast component, which are implicit conditions, are satisfied. In the second part we propose a financial application of this result: we apply the theory developed to a slow-fast local stochastic volatility model. First we prove the weak convergence of the model to a local volatility one. Then under a risk neutral measure we show that the prices of the derivatives, possibly path-dependent, converge to the ones calculated using the limit model.