Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

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

doi:10.15559/16-VMSTA66

Modern Stochastics: Theory and Applications

Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

Volume 3, Issue 4 (2016), pp. 269–285

Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili

https://doi.org/10.15559/16-VMSTA66

Pub. online: 13 December 2016 Type: Research Article

Open Access

Received
9 November 2016

Revised
3 December 2016

Accepted
4 December 2016

Published
13 December 2016

Abstract

We consider a stochastic differential equation of the form

\[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\]

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 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.

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Keywords

Stochastic differential equation weak and strong solutions stochastic volatility drift parameter estimation maximum likelihood estimator strong consistency

MSC2010

60H10 62F10 62F12

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