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

Minh-Thang Do, Hoang-Long Ngo, Dai Taguchi

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 $θ$-Euler--Maruyama scheme, which ensures that the approximated solution remains within the Weyl chamber. The second scheme is a truncated $θ$-Euler--Maruyama scheme, which produces values in $\mathbb{R}^{d}$ instead of the Weyl chamber $\mathbb{W}$, offering improved computational efficiency.

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

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 -Euler--Maruyama scheme, which ensures that the approximated solution remains within the Weyl chamber. The second scheme is a truncated -Euler--Maruyama scheme, which produces values in instead of the Weyl chamber , offering improved computational efficiency.

Paper Structure

This paper contains 10 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Radial Dunkl processes with multiplicative noise
  3. Root systems, Weyl chamber and harmonicity
  4. Existence and uniqueness of radial Dunkl processes with multiplicative noise
  5. Proof of Theorem \ref{['Thm_1']}
  6. The existence of negative moments
  7. A general theory on $\theta$-scheme for SDEs with one-sided Lipschitz drift
  8. Numerical schemes for non-colliding particle systems
  9. $\theta$-Euler--Maruyama scheme
  10. Truncated Euler--Maruyama $\theta$-scheme

Key Result

Theorem 2.2

Under Assumption Ass_1, SDE SDE_1 has a unique $\mathbb{W}$-valued strong solution. Moreover, for any $p \in (0,\infty)$, it holds that

Theorems & Definitions (23)

  • Theorem 2.2
  • Example 2.3: Radial Dunkl process
  • Example 2.4: Time-dependent Bessel process and Cox--Ingersoll--Ross model
  • Example 2.5
  • Example 2.6
  • proof : Proof of Theorem \ref{['Thm_1']}
  • Theorem 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 13 more