Strong solution and approximation of time-dependent radial Dunkl processes with multiplicative noise

TL;DR

This paper establishes strong existence and uniqueness for time-dependent radial Dunkl processes with multiplicative noise, introduces methods to analyze negative moments, and compares two numerical schemes for approximation.

Contribution

It provides new theoretical results on strong solutions and negative moments, and introduces efficient numerical schemes for these complex stochastic processes.

Findings

Proved strong existence and uniqueness of solutions.

Developed methods to establish negative moments.

Compared efficiency of two numerical approximation schemes.

Abstract

We study the strong existence and uniqueness of solutions within a Weyl chamber for a class of time-dependent particle systems driven by multiplicative noise. This class includes well-known processes in physics and mathematical finance. We propose a method to prove the existence of negative moments for the solutions. This result allows us to analyze two numerical schemes for approximating the solutions. The first scheme is a $\theta$-Euler--Maruyama scheme, which ensures that the approximated solution remains within the Weyl chamber. The second scheme is a truncated $\theta$-Euler--Maruyama scheme, which produces values in $\mathbb{R}^{d}$ instead of the Weyl chamber $\mathbb{W}$, offering improved computational efficiency.