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Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions
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 |
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. |
[2] |
R. Avikainen, On irregular functionals of SDEs and the Euler scheme,, Finance Stoch., 13 (2009), 381.
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.
doi: 10.1023/A:1022269617674. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[13] |
S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'', Academic Press, (1981).
|
[14] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'', Second edition, (2005).
|
[15] |
P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23,, Springer-Verlag, (1992).
|
[16] |
R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'', Chapman & Hall/CRC Financial Mathematics Series, (2010).
|
[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).
|
[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.
|
[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.
|
[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. |
[22] |
G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'', Scientific Computation, (2004).
|
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations,", Reprint of the second (1994) edition, (1994).
|
[24] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the second (1994) edition, (1994).
|
[25] |
A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95,, Springer-Verlag, (1996).
|
[26] |
L. Yan, The Euler scheme with irregular coefficients,, Ann. Probab., 30 (2002), 1172.
|
[27] |
H. Zähle, Weak approximation of SDEs by discrete-time processes,, J. Appl. Math. Stoch. Anal., 2008 (2757).
|
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. |
[2] |
R. Avikainen, On irregular functionals of SDEs and the Euler scheme,, Finance Stoch., 13 (2009), 381.
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.
doi: 10.1023/A:1022269617674. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[13] |
S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes,'', Academic Press, (1981).
|
[14] |
F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,'', Second edition, (2005).
|
[15] |
P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23,, Springer-Verlag, (1992).
|
[16] |
R. Korn, E. Korn and G. Kroisandt, "Monte Carlo Methods and Models in Finance and Insurance,'', Chapman & Hall/CRC Financial Mathematics Series, (2010).
|
[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).
|
[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.
|
[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.
|
[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. |
[22] |
G. N. Mil'shteĭn and M. V. Tret'yakov, "Stochastic Numerics for Mathematical Physics,'', Scientific Computation, (2004).
|
[23] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 1. Foundations,", Reprint of the second (1994) edition, (1994).
|
[24] |
L. C. G. Rogers and D. Williams, "Diffusions, Markov processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the second (1994) edition, (1994).
|
[25] |
A. N. Shiryaev, "Probability,'' Second edition, Graduate Texts in Mathematics, 95,, Springer-Verlag, (1996).
|
[26] |
L. Yan, The Euler scheme with irregular coefficients,, Ann. Probab., 30 (2002), 1172.
|
[27] |
H. Zähle, Weak approximation of SDEs by discrete-time processes,, J. Appl. Math. Stoch. Anal., 2008 (2757).
|
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