  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.
