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Strong convergence of a class of adaptive numerical methods for SDEs with jumps

Cónall Kelly, Gabriel Lord, Fandi Sun

TL;DR

This work develops Jump Adapted-Adaptive Methods (JA-AMs) for stochastic differential equations with jumps, combining jump-adapted meshes with adaptive time stepping to achieve strong $L_2$ convergence under non-global Lipschitz conditions. The framework hinges on a pair of maps, a main update $\\mathcal{M}$ and a backstop $\\B$, constrained by a mean-square one-step consistency bound and coordinated via a jump-aware mesh that includes jump times $\\tau_i$. The main theoretical result proves strong convergence of JA-AMs with rate $h_{max}^{\\delta}$, and the Jump Adapted Adaptive Milstein Method (JA-AMM) provides a concrete, order-one example (\\delta=1) under suitable moment and growth assumptions. Numerical experiments in 1D and 2D confirm order-one convergence and demonstrate competitive efficiency relative to fixed-step, jump-adapted schemes, with pronounced gains at higher jump intensities. The approach broadens the applicability of adaptive numerical methods to nonlinear SJDEs, enabling reliable simulations in finance, ecology, and other fields where jumps and nonlinearity coexist.

Abstract

We develop adaptive time-stepping strategies for Itô-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs. Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability. In this article we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case where jump and diffusion perturbations are mutually independent and the jump coefficient satisfies a global Lipschitz condition.

Strong convergence of a class of adaptive numerical methods for SDEs with jumps

TL;DR

This work develops Jump Adapted-Adaptive Methods (JA-AMs) for stochastic differential equations with jumps, combining jump-adapted meshes with adaptive time stepping to achieve strong convergence under non-global Lipschitz conditions. The framework hinges on a pair of maps, a main update and a backstop , constrained by a mean-square one-step consistency bound and coordinated via a jump-aware mesh that includes jump times . The main theoretical result proves strong convergence of JA-AMs with rate , and the Jump Adapted Adaptive Milstein Method (JA-AMM) provides a concrete, order-one example (\\delta=1) under suitable moment and growth assumptions. Numerical experiments in 1D and 2D confirm order-one convergence and demonstrate competitive efficiency relative to fixed-step, jump-adapted schemes, with pronounced gains at higher jump intensities. The approach broadens the applicability of adaptive numerical methods to nonlinear SJDEs, enabling reliable simulations in finance, ecology, and other fields where jumps and nonlinearity coexist.

Abstract

We develop adaptive time-stepping strategies for Itô-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs. Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability. In this article we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case where jump and diffusion perturbations are mutually independent and the jump coefficient satisfies a global Lipschitz condition.
Paper Structure (14 sections, 6 theorems, 74 equations, 4 figures)

This paper contains 14 sections, 6 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Finite-activity Poisson jump processes
  4. Existence and uniqueness of solutions
  5. Jump Adapted-Adaptive Methods (JA-AMs)
  6. Jump adapted meshes with procedurally generated nodes
  7. The class of Jump Adapted-Adaptive Methods (JA-AMs)
  8. The main and backstop maps
  9. Strong convergence of JA-AMs
  10. An example of a main map: the Jump Adapted Adaptive Milstein Method (JA-AMM)
  11. Numerical experiments
  12. The Monte Carlo estimator and reference solution
  13. One-dimensional test systems
  14. Two-dimensional test systems

Key Result

Lemma 3.1

For each $k\in\mathbb{N}$, if $t_k$ is an $\mathcal{F}_t$-stopping time and $h_{k+1}$ is $\mathcal{F}_{t_k}$-measurable, then $t_{k+1}$ is an $\mathcal{F}_t$-stopping time.

Figures (4)

  • Figure 1: Strong convergence and efficiency of JA-AMM for approximating the one-dimensional system with (a) and (b) for additive noise; (c) and (d) for multiplicative noise.
  • Figure 2: Strong convergence and efficiency of JA-AMM for approximating the one-dimensional system with multiplicative noise, when jump intensity increases. (a) and (b) with intensity 25; (c) and (d) with intensity 250.
  • Figure 3: Strong convergence and efficiency of JA-AMM for approximating the two-dimensional system with (a) and (b) for diagonal noise; (c) and (d) for commutative noise.
  • Figure 4: Strong convergence and efficiency of JA-AMM for approximating the two-dimensional system for non-commutative noise

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.2
  • ...and 14 more