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

TL;DR

This paper addresses the nonparametric estimation of a time-varying linear multiplier in stochastic differential equations driven by multiplicative stochastic volatility, focusing on the asymptotic behavior as noise diminishes.

Contribution

It introduces a method for estimating the unknown multiplier function in SDEs with multiplicative noise as the noise level approaches zero.

Findings

Estimation accuracy improves as noise level decreases.

The proposed estimator converges under specified conditions.

The method handles complex stochastic volatility structures.

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 $\{X_t,0\leq t \leq T\}.$