Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

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.