Estimation for the change point of the volatility in a stochastic differential equation

TL;DR

This paper develops a quasi-maximum likelihood method to detect a change point in the volatility parameter of a multidimensional Itô process, providing convergence rates and limit theorems for the estimator.

Contribution

It introduces a novel change point estimation approach for volatility in stochastic differential equations with theoretical convergence analysis.

Findings

Convergence rate of the change point estimator is established.

Limit theorems of asymptotically mixed type are proved.

Method applicable to discrete observations of multidimensional Itô processes.

Abstract

We consider a multidimensional It\^o process $Y=(Y_t)_{t\in[0,T]}$ with some unknown drift coefficient process $b_t$ and volatility coefficient $\sigma(X_t,\theta)$ with covariate process $X=(X_t)_{t\in[0,T]}$, the function $\sigma(x,\theta)$ being known up to $\theta\in\Theta$. For this model we consider a change point problem for the parameter $\theta$ in the volatility component. The change is supposed to occur at some point $t^*\in (0,T)$. Given discrete time observations from the process $(X,Y)$, we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit thereoms of aymptotically mixed type.