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Strong convergence rate of the positivity-preserving logarithmic truncated EM method for multi-dimensional stochastic differential equations with positive solutions

Xingwei Hu, Xinjie Dai, Aiguo Xiao

TL;DR

The paper addresses the problem of solving multi-dimensional SDEs with positive solutions using a positivity-preserving LTEM scheme. It extends LTEM from scalar to vector SDEs, proving a suboptimal strong convergence rate in the multidimensional setting, and shows that in the one-dimensional case the rate can be optimal by eliminating the infinitesimal factor $η(Δ)$. The main methodological contribution is the construction of a truncated logarithmic Euler–Maruyama scheme with transformation $z=\ln y$, together with a carefully designed truncation $\tilde{\lambda}_Δ,\tilde{σ}_Δ$ to ensure positivity and stability, and a novel error analysis that avoids dependence on $η(Δ)$ in 1D. Numerical experiments on stochastic Lotka–Volterra systems validate positivity preservation and reveal convergence behavior consistent with the theoretical findings. Overall, the work advances robust, positivity-preserving numerical methods for high-dimensional SDEs and clarifies how dimensionality impacts convergence rates.

Abstract

As a combination of the logarithmic transformation with the truncated Euler-Maruyama (TEM) scheme, the positivity-preserving logarithmic truncated Euler-Maruyama (LTEM) scheme has been generally developed for scalar stochastic differential equations (SDEs) with positive solutions. A subsequent question arises: can this method be extended to effectively solve general multidimensional SDEs with positive solutions? The answer to this question is affirmative. In this paper, we construct the positivity-preserving LTEM scheme to solve this type of system and demonstrate the suboptimal strong convergence rate of this scheme. On the other hand, when the underlying system degenerates into a scalar equation, the latest LTEM scheme analyzed by Tang & Mao (2024) is applicable to scalar SDEs with weak conditions, but its strong convergence rate is suboptimal. Based on this, we will theoretically demonstrate the optimal convergence rate of the LTEM method without infinitesimal factors in the scalar case. The proof strategy exactly improves its convergence rate from suboptimal to optimal. Finally, numerical examples are provided to validate the effectiveness and positivity-preserving of the LTEM method.

Strong convergence rate of the positivity-preserving logarithmic truncated EM method for multi-dimensional stochastic differential equations with positive solutions

TL;DR

The paper addresses the problem of solving multi-dimensional SDEs with positive solutions using a positivity-preserving LTEM scheme. It extends LTEM from scalar to vector SDEs, proving a suboptimal strong convergence rate in the multidimensional setting, and shows that in the one-dimensional case the rate can be optimal by eliminating the infinitesimal factor . The main methodological contribution is the construction of a truncated logarithmic Euler–Maruyama scheme with transformation , together with a carefully designed truncation to ensure positivity and stability, and a novel error analysis that avoids dependence on in 1D. Numerical experiments on stochastic Lotka–Volterra systems validate positivity preservation and reveal convergence behavior consistent with the theoretical findings. Overall, the work advances robust, positivity-preserving numerical methods for high-dimensional SDEs and clarifies how dimensionality impacts convergence rates.

Abstract

As a combination of the logarithmic transformation with the truncated Euler-Maruyama (TEM) scheme, the positivity-preserving logarithmic truncated Euler-Maruyama (LTEM) scheme has been generally developed for scalar stochastic differential equations (SDEs) with positive solutions. A subsequent question arises: can this method be extended to effectively solve general multidimensional SDEs with positive solutions? The answer to this question is affirmative. In this paper, we construct the positivity-preserving LTEM scheme to solve this type of system and demonstrate the suboptimal strong convergence rate of this scheme. On the other hand, when the underlying system degenerates into a scalar equation, the latest LTEM scheme analyzed by Tang & Mao (2024) is applicable to scalar SDEs with weak conditions, but its strong convergence rate is suboptimal. Based on this, we will theoretically demonstrate the optimal convergence rate of the LTEM method without infinitesimal factors in the scalar case. The proof strategy exactly improves its convergence rate from suboptimal to optimal. Finally, numerical examples are provided to validate the effectiveness and positivity-preserving of the LTEM method.

Paper Structure

This paper contains 12 sections, 14 theorems, 123 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multi-dimensional case and the LTEM method
  3. Notations and important lemmas
  4. The LTEM method
  5. The strong convergence analysis of the LTEM method
  6. Strong convergence rate
  7. Example and discussion
  8. One-dimensional case
  9. The LTEM mthod
  10. The optimal convergence rate of the LTEM method in one-dimensional case
  11. Numerical examples
  12. Conclusion

Key Result

lemma 1

(Lemma 2.1 in arxivhu) Let Assumption as.1 hold with the parameters satisfying $J\geq2(\alpha+1)$ and $K\geq2\beta$. Then SDE sde admits a unique global positive solution, i.e., Besides,

Figures (3)

  • Figure 1: Convergence rate
  • Figure 2: 10 trajectories of numerical solutions $y_k^1$ and $y_k^2$ generated by the truncated EM and LTEM methods for the stochastic LV model, using a step size $\Delta=2^{-6}$ and $T=2$.
  • Figure 3: Convergence rate

Theorems & Definitions (28)

  • remark 1
  • lemma 1
  • remark 2
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • corollary 1
  • ...and 18 more