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Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients

Xiaojuan Wu, Ruishu Liu, Jiahao Xu

TL;DR

This work tackles positivity preservation for numerical solutions of SDEs with non-Lipschitz coefficients in finance. By applying a Lamperti transformation and introducing an implicit term together with a correction operator, the authors construct an explicit scheme that preserves positivity and is solvable via a quadratic equation. They prove a strong convergence rate of order $1$ for transformed SDEs and demonstrate, through four key financial models (CIR, Heston-3/2, CEV, and Aït-Sahalia), that the overall method achieves the same rate after the inverse transformation, under suitable moment conditions. Numerical experiments show the scheme matches the LBEM in convergence while offering lower computational cost, highlighting its practical impact for accurate and stable simulation of finance-driven SDEs with non-Lipschitz dynamics.

Abstract

In this paper, we are interested in positivity-preserving approximations of stochastic differential equations (SDEs) with non-Lipschitz coefficients, arising from computational finance and possessing positive solutions. By leveraging a Lamperti transformation, we develop a novel, explicit, and unconditionally positivity-preserving numerical scheme for the considered financial SDEs. More precisely, an implicit term $c_{-1}Y_{n+1}^{-1}$ is incorporated in the scheme to guarantee unconditional positivity preservation, and a corrective operator is introduced in the remaining explicit terms to address the challenges posed by non-Lipschitz (possibly singular) coefficients of the transformed SDEs. By finding a unique positive root of a quadratic equation, the proposed scheme can be explicitly solved and is shown to be strongly convergent with order $1$, when used to numerically solve several well-known financial models such as the CIR process, the Heston-3/2 volatility model, the CEV process and the Aït-Sahalia model. Numerical experiments validate the theoretical findings.

Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients

TL;DR

This work tackles positivity preservation for numerical solutions of SDEs with non-Lipschitz coefficients in finance. By applying a Lamperti transformation and introducing an implicit term together with a correction operator, the authors construct an explicit scheme that preserves positivity and is solvable via a quadratic equation. They prove a strong convergence rate of order for transformed SDEs and demonstrate, through four key financial models (CIR, Heston-3/2, CEV, and Aït-Sahalia), that the overall method achieves the same rate after the inverse transformation, under suitable moment conditions. Numerical experiments show the scheme matches the LBEM in convergence while offering lower computational cost, highlighting its practical impact for accurate and stable simulation of finance-driven SDEs with non-Lipschitz dynamics.

Abstract

In this paper, we are interested in positivity-preserving approximations of stochastic differential equations (SDEs) with non-Lipschitz coefficients, arising from computational finance and possessing positive solutions. By leveraging a Lamperti transformation, we develop a novel, explicit, and unconditionally positivity-preserving numerical scheme for the considered financial SDEs. More precisely, an implicit term is incorporated in the scheme to guarantee unconditional positivity preservation, and a corrective operator is introduced in the remaining explicit terms to address the challenges posed by non-Lipschitz (possibly singular) coefficients of the transformed SDEs. By finding a unique positive root of a quadratic equation, the proposed scheme can be explicitly solved and is shown to be strongly convergent with order , when used to numerically solve several well-known financial models such as the CIR process, the Heston-3/2 volatility model, the CEV process and the Aït-Sahalia model. Numerical experiments validate the theoretical findings.

Paper Structure

This paper contains 12 sections, 15 theorems, 177 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Settings
  3. The proposed scheme and its properties
  4. Error bounds for discretizations of the transformed SDEs
  5. Applications to financial models
  6. CIR process
  7. Heston-3/2 volatility
  8. CEV Process
  9. Aït-Sahalia model
  10. Numerical Experiments
  11. Conclusion
  12. Appendix

Key Result

Lemma 2.5

Let $g$ be an FLP with ${\deg}^{-}(g) < 0 \leq {\deg}^{+}(g)$. Then $g$ is one-sided Lipschitz on $(0,+\infty)$ if and only if one of the following stands:

Figures (4)

  • Figure 1: CIR model
  • Figure 2: Heston-3/2 model
  • Figure 3: CEV model
  • Figure 4: Aït-Sahalia model

Theorems & Definitions (28)

  • Definition 2.2
  • Definition 2.3
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 18 more