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Modern Stochastics: Theory and Applications


Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility
Volume 3, Issue 4 (2016), pp. 269–285
Pub. online: 13 December 2016      Type: Research Article      Open accessOpen 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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© 2016 The Author(s). Published by VTeX
Open access article under the CC BY license.

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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