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Compound Poisson Approximation for Stochastic Volterra Equations with Singular Kernels

Xicheng Zhang, Yuanlong Zhao

TL;DR

The paper addresses strong approximation of SDEs and stochastic Volterra equations with singular kernels via a compound Poisson approach. It develops a Poisson-based discretization that avoids the need for drift continuity in time and derives explicit convergence rates: for SDEs under (H$_0$) and (H$^t_\sigma$) one has $\mathbb{E}(\sup_{t\in[0,T]}|X_t^\varepsilon-X_t|^2) \le C \varepsilon^{\gamma \wedge \beta/2}$, improving to $\varepsilon^{\gamma \wedge 1/2}$ under a special time-regularity form; for SVEs with singular kernels, the rate is $\mathbb{E}|Y_t^\varepsilon-Y_t|^2 \le C \varepsilon^{\gamma/(2(2+\gamma))}$. Theoretical results are complemented by fBM-based examples (kernel K$_H$) and numerical experiments showing enhanced stability and accuracy over Euler--Maruyama in scenarios with singular coefficients. These findings offer a robust, implementable alternative for simulations in systems with irregular time dynamics and singular kernels.

Abstract

This paper establishes the strong convergence of solutions to stochastic differential equations (SDEs) and Volterra-type SDEs when approximated by compound Poisson processes. An explicit rate of convergence is derived. A key advantage of the compound Poisson approach over the classical Euler-Maruyama method is that it does not require the drift coefficient to be continuous in the time variable and can even accommodate singularities. Numerical experiments demonstrate the stability of our approach.

Compound Poisson Approximation for Stochastic Volterra Equations with Singular Kernels

TL;DR

The paper addresses strong approximation of SDEs and stochastic Volterra equations with singular kernels via a compound Poisson approach. It develops a Poisson-based discretization that avoids the need for drift continuity in time and derives explicit convergence rates: for SDEs under (H) and (H) one has , improving to under a special time-regularity form; for SVEs with singular kernels, the rate is . Theoretical results are complemented by fBM-based examples (kernel K) and numerical experiments showing enhanced stability and accuracy over Euler--Maruyama in scenarios with singular coefficients. These findings offer a robust, implementable alternative for simulations in systems with irregular time dynamics and singular kernels.

Abstract

This paper establishes the strong convergence of solutions to stochastic differential equations (SDEs) and Volterra-type SDEs when approximated by compound Poisson processes. An explicit rate of convergence is derived. A key advantage of the compound Poisson approach over the classical Euler-Maruyama method is that it does not require the drift coefficient to be continuous in the time variable and can even accommodate singularities. Numerical experiments demonstrate the stability of our approach.

Paper Structure

This paper contains 7 sections, 18 theorems, 148 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['1']}
  3. Proof of Theorem \ref{['2']}
  4. Examples: fractional Brownian motion
  5. Numerical Experiments
  6. Linear SDEs with Singular Time-Dependent Drifts
  7. Linear Stochastic Volterra Equations with Singular Kernels

Key Result

Theorem 1.1

Assume that (H$_0$) and (H$^t_\sigma$) hold, and that the function $\ell_b$ in as_lwm satisfies, for some $\gamma \in (0,1]$, Then for any $T > 0$ and $X_0 \in L^{2}(\Omega)$, there exists a constant $C > 0$ such that for all $\varepsilon \in (0, 1)$,

Figures (2)

  • Figure 1: Graphs of $t \mapsto {\mathbb E}(X_t)$ and ${\mathbb E}|X_t|^2$.
  • Figure 2: Graphs of $t \mapsto {\mathbb E}(X_t)$ (left) and ${\mathbb E}|X_t|^2$ (right).

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 28 more