Boundary preserving explicit scheme for the Aït-Sahalia mode

Nikolaos Halidias; Ioannis S. Stamatiou

doi:10.3934/dcdsb.2022092

Article Contents

2023, Volume 28, Issue 1: 648-664. Doi: 10.3934/dcdsb.2022092

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Boundary preserving explicit scheme for the Aït-Sahalia mode

Nikolaos Halidias^1, and
Ioannis S. Stamatiou^2, ,

1.
University of the Aegean, Department of Statistics and Actuarial-Financial Mathematics, Samos, Greece
2.
University of West Attica, Department of Biomedical Sciences, Athens, Greece

^* Corresponding author: Ioannis S. Stamatiou
^* Corresponding author: Ioannis S. Stamatiou

Received: January 2022

Revised: March 2022

Early access: May 13, 2022

Published online: May 13, 2022

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Abstract

We are interested in the numerical approximation of solutions of nonlinear stochastic differential equations, that appear in financial mathematics. Here, we study the Aït-Sahalia model. We propose an explicit numerical scheme where we actually approximate the Lamperti transformation of the original stochastic differential equation and then transform back. The proposed method is domain preserving and is proven to converge strongly to the solution process with order at least $ 1 $ with no extra restrictions on the step-size $ \Delta $. Numerical experiments verify the theoretical results.

Keywords:

Explicit numerical scheme,
semi-discrete method,
non-linear stochastic differential equations,
Aït-Sahalia model.

Mathematics Subject Classification: Primary: 60H35, 65C30, 60H10, 65J15; Secondary: 65C20.

Citation:

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Figure 1. Trajectories of (9) and (35) for the approximation of (3) for different parameters with various $ \Delta $

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Figure 2. Difference between (9) and (35) for the approximation of (3) with various step-sizes

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Figure 3. Convergence of proposed method (9) for the approximation of (3) with reference solution produced at $ \Delta = 2^{-14} $

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Figure 4. Convergence of proposed method (9) for the approximation of (3) with reference solution produced at $ \Delta = 2^{-14} $ for parameter SET Ⅱ and SET Ⅲ

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Figure 5. Convergence of implicit method (35) for the approximation of (3) with reference solution produced at $ \Delta = 2^{-11} $ and $ \Delta = 2^{-10} $ for parameter SET Ⅱ and SET Ⅲ accordingly

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Figure 6. Difference between (9) and (35) for the approximation of (3) for different parameter sets

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References

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