[Paper Review] Estimating occupation time functionals

Randolf Altmeyer|arXiv (Cornell University)|Jun 11, 2017

Stochastic processes and financial applications24 references8 citations

TL;DR

This paper develops a strong $L^2$-approximation method for occupation time functionals of $d$-dimensional càdlàg processes observed discretely, relying on marginal regularity rather than Markovianity. It establishes tight upper bounds on estimation error, which are shown to be sharp (up to small polynomial factors) for Brownian motion and applied for the first time to fractional Brownian motion.

ABSTRACT

The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional cadlag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes. The results are used to approximate occupation times and local times, which is done here for fractional Brownian motion for the first time. For Brownian motion, the upper bounds are shown to be sharp, up to arbitrarily small polynomial factors.

Motivation & Objective

• To develop a robust $L^2$-approximation framework for occupation time functionals under discrete sampling of $d$-dimensional càdlàg processes.
• To generalize prior results by relaxing assumptions, particularly avoiding the need for Markovian structure.
• To provide error bounds that are applicable to non-Markovian processes, including fractional Brownian motion.
• To establish sharpness of error bounds for Brownian motion, up to arbitrarily small polynomial factors.
• To enable the first approximation of local times and occupation times for fractional Brownian motion.

Proposed method

• Utilizes regularity conditions on the finite-dimensional marginals of the process to derive error bounds.
• Applies strong $L^2$-approximation techniques to discrete observation schemes of the process.
• Derives upper bounds on the $L^2$-error of occupation time functional estimators under weak moment and continuity assumptions.
• Employs a decomposition of the occupation time functional into measurable components to control the approximation error.
• Extends the framework to non-Markovian processes by focusing on path regularity and marginal behavior rather than transition structure.
• Validates sharpness of bounds via matching lower-order polynomial factors for the case of Brownian motion.

Experimental results

Research questions

• RQ1 How can occupation time functionals be approximated under discrete observation with minimal assumptions on the underlying process?
• RQ2 What error bounds can be established for $L^2$-approximation of occupation times when the process is non-Markovian?
• RQ3 Are the derived error bounds sharp for standard diffusion processes like Brownian motion?
• RQ4 Can the proposed method be extended to self-similar processes such as fractional Brownian motion?
• RQ5 What role does marginal regularity play in controlling the approximation error for general càdlàg processes?

Key findings

• The proposed $L^2$-approximation method achieves upper bounds on estimation error under weak assumptions, generalizing prior results significantly.
• The error bounds are shown to be sharp for Brownian motion, up to arbitrarily small polynomial factors, confirming their tightness.
• The method applies to non-Markovian processes by relying on regularity of finite-dimensional marginals rather than Markovian transition laws.
• For the first time, the paper enables the approximation of local times and occupation times for fractional Brownian motion.
• The framework provides a unified approach to occupation time estimation across both Markovian and non-Markovian processes, including self-similar ones.