Nonparametric estimation of linear multiplier for stochastic differential equations driven by multiplicative stochastic volatility

We study the problem of nonparametric estimation of the linear multiplier function $\theta (t)$ for processes satisfying stochastic differential equations of the type $$d{X}_t=\theta (t)X_{t}dt+\epsilon {\sigma }_1(t,X_t){\sigma }_2(t,Y_t)d{W}_t,X_0=x_0,0\le t\le T$$ where $\{W_t,t\ge 0\}$ is a standard Brownian motion, $\{Y_t,t\ge 0\}$ is a process adapted to the filtration generated by the Brownian motion. We study the problem of estimation of the unknown function $\theta (\mathrm{.})$ as $\epsilon \to 0$ based on the observation of the process $\{X_t,0\le t\le T\}\mathrm{.}$