# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022092
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Jiang, H. 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. Szpruch, X. Mao, D. 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. Jiang, H. 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. Szpruch, X. Mao, D. 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.
Trajectories of (9) and (35) for the approximation of (3) for different parameters with various $\Delta$
Difference between (9) and (35) for the approximation of (3) with various step-sizes
Convergence of proposed method (9) for the approximation of (3) with reference solution produced at $\Delta = 2^{-14}$
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
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
Difference between (9) and (35) for the approximation of (3) for different parameter sets
 [1] Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297 [2] Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469 [3] Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761 [4] Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems and Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675 [5] Sylvie Benzoni-Gavage, Pierre Huot. Existence of semi-discrete shocks. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 163-190. doi: 10.3934/dcds.2002.8.163 [6] Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic and Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713 [7] Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104 [8] Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012 [9] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [10] Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032 [11] Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems and Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004 [12] Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021132 [13] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [14] Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3111-3133. doi: 10.3934/dcdsb.2020053 [15] Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 161-177. doi: 10.3934/dcdss.2021014 [16] Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841 [17] Tarek Saanouni. Non-linear bi-harmonic Choquard equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221 [18] Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691 [19] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control and Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015 [20] Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

2021 Impact Factor: 1.497

## Tools

Article outline

Figures and Tables