Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

We consider a stochastic differential equation of the form $$dX_t=\theta a(t,X_t)\,dt+\sigma_1(t,X_t)\sigma_2(t,Y_t)\,dW_t$$ 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 $a$, $\sigma_1$, and $\sigma_2$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter $\theta$. 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.