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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

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.

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