Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility
We consider a stochastic differential equation of the form \[dX_t=θa(t,X_t)\,dt+σ_1(t,X_t)σ_2(t,Y_t)\,dW_t\] 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 $a$, $σ_1$, and $σ_2$. 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.
math.PR↗