# American Institute of Mathematical Sciences

• Previous Article
Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's
• DCDS-B Home
• This Issue
• Next Article
Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions
July  2012, 17(5): 1455-1471. doi: 10.3934/dcdsb.2012.17.1455

## The Euler-Maruyama approximation for the absorption time of the CEV diffusion

 1 Department of Statistics, The Hebrew University, Mount Scopus, Jerusalem 91905, Israel 2 School of Mathematical Sciences, Monash University Vic 3800, Australia

Received  August 2011 Revised  January 2012 Published  March 2012

The standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme.
Citation: Pavel Chigansky, Fima C. Klebaner. The Euler-Maruyama approximation for the absorption time of the CEV diffusion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1455-1471. doi: 10.3934/dcdsb.2012.17.1455
##### References:
 [1] V. M. Abramov, F. C. Klebaner and R. Sh. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 1-14. doi: 10.3934/dcdsb.2011.16.1. [2] R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance Stoch., 13 (2009), 381-401. doi: 10.1007/s00780-009-0099-7. [3] F. Delbaen and H. Shirakawa, A note on option pricing for the constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 85-99. doi: 10.1023/A:1022269617674. [4] S. N. Ethier, Limit theorems for absorption times of genetic models, Ann. Probab., 7 (1979), 622-638. [5] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,'' Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [6] M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Comm. Statist. Simulation Comput., 28 (1999), 1135-1163. doi: 10.1080/03610919908813596. [7] M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodol. Comput. Appl. Probab., 3 (2001), 215-231. doi: 10.1023/A:1012261328124. [8] E. Gobet, Weak approximation of killed diffusion using Euler schemes, Stochastic Process. Appl., 87 (2000), 167-197. doi: 10.1016/S0304-4149(99)00109-X. [9] E. Gobet and S. Menozzi, Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme, Stochastic Process. Appl., 112 (2004), 201-223. doi: 10.1016/j.spa.2004.03.002. [10] K. Helmes, S. Röhl and R. H. Stockbridge, Computing moments of the exit time distribution for Markov processes by linear programming, Oper. Res., 49 (2001), 516-530. doi: 10.1287/opre.49.4.516.11221. [11] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes,'' Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288, Springer-Verlag, Berlin, 2003. [12] K. M. Jansons and G. D. Lythe, Exponential timestepping with boundary test for stochastic differential equations, SIAM J. Sci. Comput., 24 (2003), 1809-1822 (electronic). doi: 10.1137/S1064827501399535. [13] S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'' Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. [14] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'' Second edition, Imperial College Press, London, 2005. [15] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [16] R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'' Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL, 2010. [17] P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes, Comput. Biol. Med., 24 (1994), 91-101. [18] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales,'' Mathematics and its Applications (Soviet Series), 49, Kluwer Academic Publishers Group, Dordrecht, 1989. [19] G. N. Mil'shteĭn, Solution of the first boundary value problem for equations of parabolic type by means of the integration of stochastic differential equations, Teor. Veroyatnost. i Primenen., 40 (1995), 657-665. [20] G. N. Mil'shteĭn and M. V. Tret'yakov, The simplest random walks for the Dirichlet problem, Teor. Veroyatnost. i Primenen., 47 (2002), 39-58. [21] G. N. Mil'shteĭn and M. V. Tret'yakov, Simulation of a space-time bounded diffusion, Ann. Appl. Probab., 9 (1999), 732-779. doi: 10.1214/aoap/1029962812. [22] G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'' Scientific Computation, Springer-Verlag, Berlin, 2004. [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations," Reprint of the second (1994) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. [24] L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the second (1994) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. [25] A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1996. [26] L. Yan, The Euler scheme with irregular coefficients, Ann. Probab., 30 (2002), 1172-1194. [27] H. Zähle, Weak approximation of SDEs by discrete-time processes, J. Appl. Math. Stoch. Anal., 2008, Art. ID 275747, 15 pp.

show all references

##### References:
 [1] V. M. Abramov, F. C. Klebaner and R. Sh. Liptser, The Euler-Maruyama approximations for the CEV model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 1-14. doi: 10.3934/dcdsb.2011.16.1. [2] R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance Stoch., 13 (2009), 381-401. doi: 10.1007/s00780-009-0099-7. [3] F. Delbaen and H. Shirakawa, A note on option pricing for the constant elasticity of variance model, Asia-Pacific Financial Markets, 9 (2002), 85-99. doi: 10.1023/A:1022269617674. [4] S. N. Ethier, Limit theorems for absorption times of genetic models, Ann. Probab., 7 (1979), 622-638. [5] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,'' Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [6] M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Comm. Statist. Simulation Comput., 28 (1999), 1135-1163. doi: 10.1080/03610919908813596. [7] M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodol. Comput. Appl. Probab., 3 (2001), 215-231. doi: 10.1023/A:1012261328124. [8] E. Gobet, Weak approximation of killed diffusion using Euler schemes, Stochastic Process. Appl., 87 (2000), 167-197. doi: 10.1016/S0304-4149(99)00109-X. [9] E. Gobet and S. Menozzi, Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme, Stochastic Process. Appl., 112 (2004), 201-223. doi: 10.1016/j.spa.2004.03.002. [10] K. Helmes, S. Röhl and R. H. Stockbridge, Computing moments of the exit time distribution for Markov processes by linear programming, Oper. Res., 49 (2001), 516-530. doi: 10.1287/opre.49.4.516.11221. [11] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes,'' Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288, Springer-Verlag, Berlin, 2003. [12] K. M. Jansons and G. D. Lythe, Exponential timestepping with boundary test for stochastic differential equations, SIAM J. Sci. Comput., 24 (2003), 1809-1822 (electronic). doi: 10.1137/S1064827501399535. [13] S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'' Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. [14] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'' Second edition, Imperial College Press, London, 2005. [15] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [16] R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'' Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL, 2010. [17] P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes, Comput. Biol. Med., 24 (1994), 91-101. [18] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales,'' Mathematics and its Applications (Soviet Series), 49, Kluwer Academic Publishers Group, Dordrecht, 1989. [19] G. N. Mil'shteĭn, Solution of the first boundary value problem for equations of parabolic type by means of the integration of stochastic differential equations, Teor. Veroyatnost. i Primenen., 40 (1995), 657-665. [20] G. N. Mil'shteĭn and M. V. Tret'yakov, The simplest random walks for the Dirichlet problem, Teor. Veroyatnost. i Primenen., 47 (2002), 39-58. [21] G. N. Mil'shteĭn and M. V. Tret'yakov, Simulation of a space-time bounded diffusion, Ann. Appl. Probab., 9 (1999), 732-779. doi: 10.1214/aoap/1029962812. [22] G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'' Scientific Computation, Springer-Verlag, Berlin, 2004. [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations," Reprint of the second (1994) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. [24] L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus," Reprint of the second (1994) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. [25] A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1996. [26] L. Yan, The Euler scheme with irregular coefficients, Ann. Probab., 30 (2002), 1172-1194. [27] H. Zähle, Weak approximation of SDEs by discrete-time processes, J. Appl. Math. Stoch. Anal., 2008, Art. ID 275747, 15 pp.
 [1] Vyacheslav M. Abramov, Fima C. Klebaner, Robert Sh. Lipster. The Euler-Maruyama approximations for the CEV model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 1-14. doi: 10.3934/dcdsb.2011.16.1 [2] Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198 [3] Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 [4] Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 [5] Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure and Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031 [6] Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481 [7] Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393 [8] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [9] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 [10] Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092 [11] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [12] Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479 [13] Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic and Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 [14] Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 [15] Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 [16] Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823 [17] Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic and Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991 [18] Rinaldo M. Colombo, Francesca Marcellini. Coupling conditions for the $3\times 3$ Euler system. Networks and Heterogeneous Media, 2010, 5 (4) : 675-690. doi: 10.3934/nhm.2010.5.675 [19] Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885 [20] Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4231-4253. doi: 10.3934/dcdsb.2021225

2021 Impact Factor: 1.497