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Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Ait-Sahalia model

Ruishu Liu, Yulin Cao, Xiaojie Wang

Abstract

The present work is devoted to strong approximations of a generalized Aït-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size $h >0$. A mean-square convergence rate of order $0.5$ is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.

Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Ait-Sahalia model

Abstract

The present work is devoted to strong approximations of a generalized Aït-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size . A mean-square convergence rate of order is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.
Paper Structure (6 sections, 11 theorems, 95 equations, 3 figures, 2 tables)

This paper contains 6 sections, 11 theorems, 95 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The generalized Aït-Sahalia model
  3. The explicit positivity-preserving Euler-type scheme
  4. Mean-square convergence rate of the proposed scheme
  5. Numerical experiments
  6. Conclusion

Key Result

Theorem 2.1

Let $c_{-1},c_{0},c_{1},c_{2},c_{3}>0$, $\kappa$, $\rho>1$. Given any initial value $X_0 > 0$, there exists a unique, positive global solution $X_t$ to the SDE 2023AS-eq:model_SDE on $t \geq 0$.

Figures (3)

  • Figure 1: Strong convergence rate of TEM and BEM for Example \ref{['2023AS-eg1']} (left) and Example \ref{['2023AS-eg2']} (right).
  • Figure 2: Strong convergence rate of TEM and BEM for Example \ref{['2023AS-eg3']}.
  • Figure 3: Fifty paths of numerical solutions for BEM and TEM

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1: Unconditional positivity-preserving
  • Example 3.5
  • Lemma 3.6
  • Corollary 3.7
  • ...and 5 more