Overview
This research focuses on the problem of drift estimation for rough processes within a small noise asymptotic framework. It introduces a Trajectory Fitting Estimator designed to determine an unknown parameter, $\theta^\star$, embedded within the drift function of a stochastic Volterra equation. The study characterizes the properties of this estimator, specifically its consistency and asymptotic normality, under conditions involving singular Volterra kernels and a diffusion coefficient attenuated by a small noise parameter, $\ve \to 0$. Additionally, the research specifies conditions necessary for the $L^p$ convergence of the estimator.
Research Context
The core subject of this investigation is a process, denoted as $X^\ve$, which is defined as the solution to a stochastic Volterra equation. This equation incorporates an unknown parameter, $\theta^\star$, within its drift function. A key characteristic of the processes considered is the nature of the Volterra kernel, which is specified as singular. An illustrative example of such a kernel is given as $K_0(u)=c u^{\alpha-1/2} \id{u > 0}$, where the parameter $\alpha$ is within the range $(0,1/2)$. A further defining aspect of this research context is the assumption that the diffusion coefficient associated with the process $X^\ve$ scales proportionally with a small parameter $\ve$, which tends towards zero.
Approach
The methodology proposed in this research centers on the construction of a Trajectory Fitting Estimator. This estimator is developed based on observations of the path of the process $X^\ve$ over a continuous interval, specifically $(X^\ve_s)_{s\in[0,T]}$. The objective of this estimator is to determine the unknown parameter $\theta^\star$ present in the drift function of the underlying stochastic Volterra equation.
Findings
- The Trajectory Fitting Estimator, constructed from observations of the process path $(X^\ve_s)_{s\in[0,T]}$, has been shown to be consistent.
- The estimator also exhibits asymptotic normality.
- Identifiability conditions were specified, which are necessary to ensure the $L^p$ convergence of the estimator.