Numerical Approximation of Stochastic Differential Equations Driven by Fractional Brownian Motion for All Hurst Parameter Values

Core Concepts

This paper presents a numerical method for efficiently approximating the solution of stochastic differential equations driven by fractional Brownian motion for all values of the Hurst parameter H in the range (0,1). The method is based on the Wick-Itô-Skorohod (WIS) integral, which is well-defined and centered for all H.

Abstract

The paper examines the numerical approximation of a quasilinear stochastic differential equation (SDE) with multiplicative fractional Brownian motion. The stochastic integral is interpreted in the Wick-Itô-Skorohod (WIS) sense, which is well-defined and centered for all H ∈ (0,1).
The key highlights and insights are:

The authors provide an introduction to the theory of WIS integration before examining the existence and uniqueness of a solution to the SDE.

They introduce a numerical method, GBMEM, based on theoretical results in prior work, and construct a translation operator required for the practical implementation of the method.

A strong convergence result is proved, showing an error of O(Δt^H) in the non-autonomous case and O(Δt^min(H,ζ)) in the non-autonomous case, where ζ is a Hölder continuity parameter.

Numerical experiments are presented, and the authors conjecture that the theoretical results may not be optimal, as they observe numerically a rate of min(H+1/2, 1) in the autonomous case.

This work opens up the possibility to efficiently simulate SDEs for all H values, including small values of H when the stochastic integral is interpreted in the WIS sense.

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Key Insights Distilled From

Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration

by Utku Erdogan... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07013.pdf

Deeper Inquiries

How can the theoretical results be improved to match the observed faster convergence rates in the numerical experiments

To improve the theoretical results to match the observed faster convergence rates in the numerical experiments, several approaches can be considered. One option is to revisit the assumptions made in the theoretical analysis and see if they can be relaxed without compromising the validity of the results. This could involve exploring different conditions on the coefficients of the SDE or the properties of the noise process. Additionally, refining the numerical method used in the experiments and incorporating more advanced techniques could help bridge the gap between theory and practice. By refining the theoretical framework to better align with the observed convergence rates, a more accurate and comprehensive understanding of the problem can be achieved.

What are the potential applications of efficiently simulating SDEs driven by fractional Brownian motion for all Hurst parameter values

Efficiently simulating SDEs driven by fractional Brownian motion for all Hurst parameter values has a wide range of potential applications across various fields. In finance, these simulations can be used for pricing financial derivatives, risk management, and portfolio optimization. In physics, they can be applied to model complex systems with long-range dependencies, such as turbulent flows or particle diffusion. In biology, SDEs driven by fractional Brownian motion can be used to model genetic processes, population dynamics, and ecological systems. The ability to simulate SDEs for all Hurst parameter values provides a versatile tool for studying and understanding a diverse range of phenomena characterized by non-Markovian behavior.

Can the WIS integration framework be extended to other types of non-Markovian noise processes beyond fractional Brownian motion

The Wick-Itˆo-Skorohod (WIS) integration framework used for SDEs driven by fractional Brownian motion can potentially be extended to other types of non-Markovian noise processes beyond fractional Brownian motion. By adapting the concepts and techniques developed for fractional Brownian motion, similar integration methods could be developed for other non-Markovian processes, such as fractional Gaussian noise, Lévy processes, or other types of long-memory processes. This extension would require a careful analysis of the properties of the specific noise process in question and the development of appropriate integration techniques within the WIS framework. The flexibility and robustness of the WIS integration approach make it a promising candidate for extending to a broader class of non-Markovian noise processes.