Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

Meriem Bel Hadj Khlifa; Yuliya Mishura; Kostiantyn Ralchenko; Mounir Zili

doi:10.15559/16-VMSTA66

paper

Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

arXiv:1701.01238 · doi:10.15559/16-VMSTA66

Abstract

We consider a stochastic differential equation of the form \[dX_t=θa(t,X_t)\,dt+σ_1(t,X_t)σ_2(t,Y_t)\,dW_t\] 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 $a$ , $σ_{1}$ , and $σ_{2}$ . Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter $θ$ . 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.

Published at http://dx.doi.org/10.15559/16-VMSTA66 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)