Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

TL;DR

This paper investigates parameter estimation in a stochastic differential equation with multiplicative stochastic volatility, establishing existence, uniqueness, and the strong consistency of the maximum likelihood estimator under various conditions.

Contribution

It provides new theoretical results on the existence, uniqueness, and consistency of estimators for SDEs with multiplicative stochastic volatility where the volatility process follows its own diffusion.

Findings

Proved existence and uniqueness of solutions under various conditions.

Established strong consistency of the maximum likelihood estimator.

Provided examples demonstrating the theoretical results.

Abstract

We consider a stochastic differential equation of the form \[dX_t=\theta a(t,X_t)\,dt+\sigma_1(t,X_t)\sigma_2(t,Y_t)\,dW_t\] with multiplicative stochastic volatility, where $Y$ is some adapted stochastic process. We prove existence--uniqueness results for weak and strong solutions of this equation under various conditions on the process $Y$ and the coefficients $a$, $\sigma_1$, and $\sigma_2$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter $\theta$. We suppose that $Y$ is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process $Y$ are provided supplying the strong consistency.