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Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients

Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao

Abstract

In this paper, we propose two variants of the positivity-preserving schemes, namely the truncated Euler-Maruyama (EM) method and the truncated Milstein scheme, applied to stochastic differential equations (SDEs) with positive solutions and super-linear coefficients. Under some regularity and integrability assumptions we derive the optimal strong convergence rates of the two schemes. Moreover, we demonstrate flexibility of our approaches by applying the truncated methods to approximate SDEs with super-linear coefficients (3/2 and Aiıt-Sahalia models) directly and also with sub-linear coefficients (CIR model) indirectly. Numerical experiments are provided to verify the effectiveness of the theoretical results.

Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients

Abstract

In this paper, we propose two variants of the positivity-preserving schemes, namely the truncated Euler-Maruyama (EM) method and the truncated Milstein scheme, applied to stochastic differential equations (SDEs) with positive solutions and super-linear coefficients. Under some regularity and integrability assumptions we derive the optimal strong convergence rates of the two schemes. Moreover, we demonstrate flexibility of our approaches by applying the truncated methods to approximate SDEs with super-linear coefficients (3/2 and Aiıt-Sahalia models) directly and also with sub-linear coefficients (CIR model) indirectly. Numerical experiments are provided to verify the effectiveness of the theoretical results.
Paper Structure (14 sections, 18 theorems, 174 equations, 2 figures, 3 tables)

This paper contains 14 sections, 18 theorems, 174 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setting and preliminaries
  3. Main Results
  4. Positivity-preserving Truncated EM (TEM) scheme
  5. Positivity-preserving Truncated Milstein (TMil) scheme
  6. Applications
  7. 3/2 model
  8. Aït-Sahalia model
  9. CIR model
  10. Numerical experiments
  11. Appendix
  12. Proof of Lemma \ref{['Lem1']}
  13. Proof of Lemma \ref{['Lem3.3_2']}
  14. Proof of Lemma \ref{['Lem3.1']}

Key Result

Lemma 2.4

Let Assumption Assu1 hold. Let $\pi_{\Delta}$, $f_{\Delta}$ and $g_{\Delta}$ be the truncated mapping and functions defined in eq23, eq2 respectively. Then

Figures (2)

  • Figure 1: Convergence rates for 3/2 and AIT models
  • Figure 2: Sample trajectories computed with different schemes for AIT

Theorems & Definitions (24)

  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Theorem 3.2: Convergence order of TEM
  • Remark 3.3
  • Proposition 3.4: Moment boundedness of TEM
  • Corollary 3.5
  • Remark 3.6
  • Lemma 3.8
  • ...and 14 more