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Boundary-preserving Lamperti-splitting schemes for some Stochastic Differential Equations

Johan Ulander

TL;DR

The paper develops boundary-preserving numerical schemes for scalar Itô SDEs with bounded state-spaces by applying a Lamperti transform to obtain additive diffusion and then a Lie–Trotter splitting. It proves $L^{p}(\Omega)$-convergence of order $1$ for all $p\ge1$ for both a semi-analytic LS scheme and a fully discrete LS scheme (the latter with an inexact ODE solve under a mild time-step restriction), while ensuring the numerical solution remains inside the interior of the domain. The approach yields explicit representation formulas and accommodates non-globally Lipschitz drifts and diffusions with potential superlinear growth. Numerical experiments on SIS, Nagumo, and Allen–Cahn type SDEs demonstrate boundary preservation and convergence rates, outperforming standard schemes in terms of domain confinement. Overall, the Lamperti–splitting framework offers a robust, boundary-respecting alternative for strong approximation of bounded-domain SDEs with non-Lipschitz coefficients.

Abstract

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(Ω)$-convergence of order $1$, for every $p \geq 1$, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

Boundary-preserving Lamperti-splitting schemes for some Stochastic Differential Equations

TL;DR

The paper develops boundary-preserving numerical schemes for scalar Itô SDEs with bounded state-spaces by applying a Lamperti transform to obtain additive diffusion and then a Lie–Trotter splitting. It proves -convergence of order for all for both a semi-analytic LS scheme and a fully discrete LS scheme (the latter with an inexact ODE solve under a mild time-step restriction), while ensuring the numerical solution remains inside the interior of the domain. The approach yields explicit representation formulas and accommodates non-globally Lipschitz drifts and diffusions with potential superlinear growth. Numerical experiments on SIS, Nagumo, and Allen–Cahn type SDEs demonstrate boundary preservation and convergence rates, outperforming standard schemes in terms of domain confinement. Overall, the Lamperti–splitting framework offers a robust, boundary-respecting alternative for strong approximation of bounded-domain SDEs with non-Lipschitz coefficients.

Abstract

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove -convergence of order , for every , of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.
Paper Structure (16 sections, 10 theorems, 124 equations, 6 figures, 3 tables)

This paper contains 16 sections, 10 theorems, 124 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Description of the SDE
  4. Boundary classification
  5. A semi-analytic boundary-preserving integrator
  6. Description of the integrator
  7. Convergence result
  8. A boundary-preserving integrator
  9. Description of the integrator
  10. Convergence result
  11. Numerical experiments
  12. Proof of a lemma
  13. Additional formulas
  14. SIS SDE
  15. Nagumo SDE
  16. ...and 1 more sections

Key Result

Proposition 1

Suppose Assumption ass:g is satisfied. Then $\Phi^{-1}: \mathbb{R} \to \mathring{D}$ is bounded, bijective, continuously differentiable and has bounded derivative. In particular, $\Phi^{-1}: \mathbb{R} \to \mathring{D}$ is globally Lipschitz continuous and we denote the Lipschitz constant of $\Phi^{

Figures (6)

  • Figure 1: Path comparison of the EM, SEM, TE and LS schemes applied to the SIS SDE with parameters $\lambda = 4$, $x_{0} = 0.9$, $T = 0.4$ and $M=50$.
  • Figure 2: $L^{2}(\Omega)$-errors on the interval $[0,1]$ of the Lamperti-splitting scheme (LS) for the SIS SDE for different choices of $\lambda>0$ and reference lines with slopes $1/2$ and $1$. Averaged over $300$ samples.
  • Figure 3: Path comparison of the EM, SEM, TEM and LS schemes applied to the Nagumo SDE with parameters $\lambda = 4$, $x_{0}=0.9$, $T = 0.4$ and $M=50$.
  • Figure 4: $L^{2}(\Omega)$-errors on the interval $[0,1]$ of the Lamperti-splitting scheme (LS) for the Nagumo SDE for different choices of $\lambda>0$ and reference lines with slopes $1/2$ and $1$. Averaged over $300$ samples.
  • Figure 5: Path comparison of the EM, SEM, TE and LS schemes applied to the Allen-Cahn SDE with parameters $\lambda = 3$, $x_{0}=0.9$, $T = 0.4$ and $M=50$.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Proposition 1
  • proof : Proof of Proposition \ref{['prop:phiinv']}
  • Proposition 2
  • proof : Proof of Proposition \ref{['prop:H']}
  • Remark 3
  • Proposition 4
  • proof : Proof of Proposition \ref{['propo:BP-scheme']}
  • Remark 5
  • Remark 6
  • Theorem 7
  • ...and 13 more