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 $\hat{I}¸(t)$ for processes satisfying stochastic differential equations of the type $$d{X}_t=\hat{I}¸(t)X_{t}dt+\hat{I}\mu \ddot{I}{ƒ}_1(t,X_t)\ddot{I}{ƒ}_2(t,Y_t)d{W}_t,X_0=x_0,0\le t\le T$$ 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 $\hat{I}¸(.)$ as $\hat{I}\mu \to 0$ based on the observation of the process $\{X_t,0\le t\le T\}.$