Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Explicit Euler-type Scheme for Lévy-driven SDEs with Superlinear and Time-Irregular Coefficients

Sani Biswas, Joaquin Fontbona

TL;DR

The paper addresses strong numerical approximation of Lévy-driven SDEs with superlinear and time-irregular drift (Carathéodory type). It introduces a randomized tamed Euler scheme that combines drift taming with randomized evaluation times to cope with time irregularity and jumps. The main theoretical contributions are a strong $L^2$-convergence rate arbitrarily close to $0.5$ under mild time-regularity assumptions, and an extension to Lévy-driven stochastic delay differential equations with Markovian switching, with explicit taming forms verified to satisfy the required conditions. Numerical experiments on Levy-driven double-well dynamics corroborate the theory and illustrate applicability to complex jump-diffusion models.

Abstract

This paper introduces a randomized tamed Euler scheme tailored for Lévy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carathéodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong $\mathcal{L}^2$-convergence rate, arbitrarily close to $0.5$. A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for Lévy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carathéodory-type SDEs and even for ordinary differential equations.

An Explicit Euler-type Scheme for Lévy-driven SDEs with Superlinear and Time-Irregular Coefficients

TL;DR

The paper addresses strong numerical approximation of Lévy-driven SDEs with superlinear and time-irregular drift (Carathéodory type). It introduces a randomized tamed Euler scheme that combines drift taming with randomized evaluation times to cope with time irregularity and jumps. The main theoretical contributions are a strong -convergence rate arbitrarily close to under mild time-regularity assumptions, and an extension to Lévy-driven stochastic delay differential equations with Markovian switching, with explicit taming forms verified to satisfy the required conditions. Numerical experiments on Levy-driven double-well dynamics corroborate the theory and illustrate applicability to complex jump-diffusion models.

Abstract

This paper introduces a randomized tamed Euler scheme tailored for Lévy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carathéodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong -convergence rate, arbitrarily close to . A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for Lévy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carathéodory-type SDEs and even for ordinary differential equations.
Paper Structure (11 sections, 13 theorems, 75 equations, 1 figure, 1 table)

This paper contains 11 sections, 13 theorems, 75 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Required Preliminaries and Elaboration of Key Findings
  3. Well-posedness and Moment Bounds
  4. Randomized Tamed Euler Scheme and Main Result
  5. Suggested Formulation of Taming
  6. Randomized Tamed Euler Scheme for Lévy-driven SDDE with Markovian Switching
  7. Required Preliminaries
  8. Moment bound of the randomized tamed Euler scheme
  9. Rate of Convergence
  10. Numerical Experiment
  11. Verifications of Assumptions

Key Result

Proposition 2.1

If Assumptions asum:monotonocity to asum:continuity are met, the SDE eq:sde has a unique solution. . Further, if Assumptions asum:ic and asum:coercivity_p are also true, there exists a constant $K>0$, such that $\sup_{t\in[0,T]}E|x_t|^{q}\leq K.$

Figures (1)

  • Figure 1: $\mathcal{L}^p$-error of randomized tamed Euler scheme \ref{['eq:scm']} for SDE \ref{['eq:DWD']}.

Theorems & Definitions (25)

  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.4
  • Lemma 3.1
  • ...and 15 more