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

Abstract: We study the problem of nonparametric estimation of the linear multiplier function $\theta(t)$ for processes satisfying stochastic differential equations of the type $$dX_t= \theta(t)X_t dt+ \epsilon\; \sigma_1(t,X_t)\sigma_2(t,Y_t)dW_t, X_0=x_0, 0 \leq t \leq T$$ where ${W_t, t\geq 0}$ is a standard Brownian motion, ${Y_t, t\geq 0}$ is a process adapted to the filtration generated by the Brownian motion. We study the problem of estimation of the unknown function $\theta(.)$ as $\epsilon \rightarrow 0$ based on the observation of the process ${X_t,0\leq t \leq T}.$