doi: 10.3934/dcdsb.2022092
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Boundary preserving explicit scheme for the Aït-Sahalia mode

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

Received  January 2022 Revised  March 2022 Early access May 2022

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.

Citation: Nikolaos Halidias, Ioannis S. Stamatiou. Boundary preserving explicit scheme for the Aït-Sahalia mode. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022092
References:
[1]

Y. Aït-Sahalia, Testing continuous-time models of the spot interest rate, The Review of Financial Studies, 9 (1996), 385-426.  doi: 10.1093/rfs/9.2.385.

[2]

N. Halidias, Semi-discrete approximations for stochastic differential equations and applications, Int. J. Comput. Math., 89 (2012), 780-794.  doi: 10.1080/00207160.2012.658380.

[3]

N. Halidias, On the construction of boundary preserving numerical schemes, Monte Carlo Methods Appl., 22 (2016), 277-289.  doi: 10.1515/mcma-2016-0113.

[4]

F. JiangH. Yang and T. Tian, Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 101-113.  doi: 10.3934/DCDSB.2017005.

[5]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, 1$^st$ edition, Springer-Verlag Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-57913-4.

[6]

A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numer. Math., 128 (2014), 103-136.  doi: 10.1007/s00211-014-0606-4.

[7]

I. S. Stamatiou, The semi-discrete method for the approximation of the solution of stochastic differential equations, in Nonlinear Analysis, Differential Equations, and Applications (eds. T.M. Rassias), Springer, Cham, (2021), 625–638. doi: 10.1007/978-3-030-72563-1_23.

[8]

L. SzpruchX. MaoD. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numerical Mathematics, 51 (2011), 405-425.  doi: 10.1007/s10543-010-0288-y.

[9]

D. Willett and J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall's inequality, Monatsh. Math., 69 (1965), 362-367.  doi: 10.1007/BF01297622.

show all references

References:
[1]

Y. Aït-Sahalia, Testing continuous-time models of the spot interest rate, The Review of Financial Studies, 9 (1996), 385-426.  doi: 10.1093/rfs/9.2.385.

[2]

N. Halidias, Semi-discrete approximations for stochastic differential equations and applications, Int. J. Comput. Math., 89 (2012), 780-794.  doi: 10.1080/00207160.2012.658380.

[3]

N. Halidias, On the construction of boundary preserving numerical schemes, Monte Carlo Methods Appl., 22 (2016), 277-289.  doi: 10.1515/mcma-2016-0113.

[4]

F. JiangH. Yang and T. Tian, Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 101-113.  doi: 10.3934/DCDSB.2017005.

[5]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, 1$^st$ edition, Springer-Verlag Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-57913-4.

[6]

A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numer. Math., 128 (2014), 103-136.  doi: 10.1007/s00211-014-0606-4.

[7]

I. S. Stamatiou, The semi-discrete method for the approximation of the solution of stochastic differential equations, in Nonlinear Analysis, Differential Equations, and Applications (eds. T.M. Rassias), Springer, Cham, (2021), 625–638. doi: 10.1007/978-3-030-72563-1_23.

[8]

L. SzpruchX. MaoD. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numerical Mathematics, 51 (2011), 405-425.  doi: 10.1007/s10543-010-0288-y.

[9]

D. Willett and J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall's inequality, Monatsh. Math., 69 (1965), 362-367.  doi: 10.1007/BF01297622.

Figure 1.  Trajectories of (9) and (35) for the approximation of (3) for different parameters with various $ \Delta $
Figure 2.  Difference between (9) and (35) for the approximation of (3) with various step-sizes
Figure 3.  Convergence of proposed method (9) for the approximation of (3) with reference solution produced at $ \Delta = 2^{-14} $
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
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
Figure 6.  Difference between (9) and (35) for the approximation of (3) for different parameter sets
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