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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

Pavel Chigansky 1, and  Fima C. Klebaner 2,

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.
Keywords: absorption times, weak convergence, Euler-Maruyama scheme, Diffusion, CEV model..
Mathematics Subject Classification: Primary: 60H10, 60H35; Secondary: 60F17, 91G6.
Citation: Pavel Chigansky, Fima C. Klebaner. The Euler-Maruyama approximation for the absorption time of the CEV diffusion. Discrete & 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.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar

[2]

R. Avikainen, On irregular functionals of SDEs and the Euler scheme,, Finance Stoch., 13 (2009), 381.  doi: 10.1007/s00780-009-0099-7.  Google Scholar

[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.  doi: 10.1023/A:1022269617674.  Google Scholar

[4]

S. N. Ethier, Limit theorems for absorption times of genetic models,, Ann. Probab., 7 (1979), 622.   Google Scholar

[5]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,'', Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986).   Google Scholar

[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.  doi: 10.1080/03610919908813596.  Google Scholar

[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.  doi: 10.1023/A:1012261328124.  Google Scholar

[8]

E. Gobet, Weak approximation of killed diffusion using Euler schemes,, Stochastic Process. Appl., 87 (2000), 167.  doi: 10.1016/S0304-4149(99)00109-X.  Google Scholar

[9]

E. Gobet and S. Menozzi, Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme,, Stochastic Process. Appl., 112 (2004), 201.  doi: 10.1016/j.spa.2004.03.002.  Google Scholar

[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.  doi: 10.1287/opre.49.4.516.11221.  Google Scholar

[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, (2003).   Google Scholar

[12]

K. M. Jansons and G. D. Lythe, Exponential timestepping with boundary test for stochastic differential equations,, SIAM J. Sci. Comput., 24 (2003), 1809.  doi: 10.1137/S1064827501399535.  Google Scholar

[13]

S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'', Academic Press, (1981).   Google Scholar

[14]

F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'', Second edition, (2005).   Google Scholar

[15]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23,, Springer-Verlag, (1992).   Google Scholar

[16]

R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'', Chapman & Hall/CRC Financial Mathematics Series, (2010).   Google Scholar

[17]

P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes,, Comput. Biol. Med., 24 (1994), 91.   Google Scholar

[18]

R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales,'' Mathematics and its Applications (Soviet Series), 49,, Kluwer Academic Publishers Group, (1989).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[21]

G. N. Mil'shteĭn and M. V. Tret'yakov, Simulation of a space-time bounded diffusion,, Ann. Appl. Probab., 9 (1999), 732.  doi: 10.1214/aoap/1029962812.  Google Scholar

[22]

G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'', Scientific Computation, (2004).   Google Scholar

[23]

L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations,", Reprint of the second (1994) edition, (1994).   Google Scholar

[24]

L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the second (1994) edition, (1994).   Google Scholar

[25]

A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95,, Springer-Verlag, (1996).   Google Scholar

[26]

L. Yan, The Euler scheme with irregular coefficients,, Ann. Probab., 30 (2002), 1172.   Google Scholar

[27]

H. Zähle, Weak approximation of SDEs by discrete-time processes,, J. Appl. Math. Stoch. Anal., 2008 (2757).   Google Scholar

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.  doi: 10.3934/dcdsb.2011.16.1.  Google Scholar

[2]

R. Avikainen, On irregular functionals of SDEs and the Euler scheme,, Finance Stoch., 13 (2009), 381.  doi: 10.1007/s00780-009-0099-7.  Google Scholar

[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.  doi: 10.1023/A:1022269617674.  Google Scholar

[4]

S. N. Ethier, Limit theorems for absorption times of genetic models,, Ann. Probab., 7 (1979), 622.   Google Scholar

[5]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,'', Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986).   Google Scholar

[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.  doi: 10.1080/03610919908813596.  Google Scholar

[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.  doi: 10.1023/A:1012261328124.  Google Scholar

[8]

E. Gobet, Weak approximation of killed diffusion using Euler schemes,, Stochastic Process. Appl., 87 (2000), 167.  doi: 10.1016/S0304-4149(99)00109-X.  Google Scholar

[9]

E. Gobet and S. Menozzi, Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme,, Stochastic Process. Appl., 112 (2004), 201.  doi: 10.1016/j.spa.2004.03.002.  Google Scholar

[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.  doi: 10.1287/opre.49.4.516.11221.  Google Scholar

[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, (2003).   Google Scholar

[12]

K. M. Jansons and G. D. Lythe, Exponential timestepping with boundary test for stochastic differential equations,, SIAM J. Sci. Comput., 24 (2003), 1809.  doi: 10.1137/S1064827501399535.  Google Scholar

[13]

S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'', Academic Press, (1981).   Google Scholar

[14]

F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'', Second edition, (2005).   Google Scholar

[15]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23,, Springer-Verlag, (1992).   Google Scholar

[16]

R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'', Chapman & Hall/CRC Financial Mathematics Series, (2010).   Google Scholar

[17]

P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes,, Comput. Biol. Med., 24 (1994), 91.   Google Scholar

[18]

R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales,'' Mathematics and its Applications (Soviet Series), 49,, Kluwer Academic Publishers Group, (1989).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[21]

G. N. Mil'shteĭn and M. V. Tret'yakov, Simulation of a space-time bounded diffusion,, Ann. Appl. Probab., 9 (1999), 732.  doi: 10.1214/aoap/1029962812.  Google Scholar

[22]

G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'', Scientific Computation, (2004).   Google Scholar

[23]

L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations,", Reprint of the second (1994) edition, (1994).   Google Scholar

[24]

L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the second (1994) edition, (1994).   Google Scholar

[25]

A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95,, Springer-Verlag, (1996).   Google Scholar

[26]

L. Yan, The Euler scheme with irregular coefficients,, Ann. Probab., 30 (2002), 1172.   Google Scholar

[27]

H. Zähle, Weak approximation of SDEs by discrete-time processes,, J. Appl. Math. Stoch. Anal., 2008 (2757).   Google Scholar

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