Nonparametric estimation for a stochastic volatility model

TL;DR

This paper introduces nonparametric least squares estimators for the drift and diffusion coefficients of an unobserved diffusion process in a stochastic volatility model, with theoretical risk bounds and data-driven model selection, demonstrated through simulations.

Contribution

It develops novel nonparametric estimation methods for unobserved diffusion processes in stochastic volatility models, including risk bounds and adaptive dimension selection.

Findings

Estimators achieve bounded risk under certain conditions.

Simulation results demonstrate the effectiveness of the proposed method.

Data-driven procedure successfully selects model complexity.

Abstract

Consider discrete time observations (X_{\ell\delta})_{1\leq \ell \leq n+1}$ of the process $X$ satisfying $dX_t= \sqrt{V_t} dB_t$, with $V_t$ a one-dimensional positive diffusion process independent of the Brownian motion $B$. For both the drift and the diffusion coefficient of the unobserved diffusion $V$, we propose nonparametric least square estimators, and provide bounds for theirrisk. Estimators are chosen among a collection of functions belonging to a finite dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.