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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), 1-14.
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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