An estimate of the rate of convergence of an approximating scheme applied to a stochastic differential equation with an additional parameter
Authors:
Yu. S. Mishura and A. V. Shvaĭ
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 82 (2011), 75-85
MSC (2010):
Primary 60H10
DOI:
https://doi.org/10.1090/S0094-9000-2011-00828-9
Published electronically:
August 4, 2011
MathSciNet review:
2790482
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider a stochastic differential equation with a diffusion coefficient involving an additional process viewed as a parameter. Given a rate of convergence of the Euler approximations for this parameter, we find the mean square rate of convergence of the Euler approximation scheme. An example is considered where the parameter is driven by a fractional Brownian motion.
References
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- D. F. Kuznetsov, Stokhasticheskie differentsial′nye uravneniya: teoriya i praktika chislennogo resheniya, 2nd ed., Sankt-Peterburgskiĭ Gosudarstvennyĭ Politekhnicheskiĭ Universitet, St. Petersburg, 2007. MR 2510680
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
- G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR 1335454
- V. S. Korolyuk and N. Limnios, Poisson approximation of increment processes with Markov switching, Teor. Veroyatn. Primen. 49 (2004), no. 4, 726–742 (English, with Russian summary); English transl., Theory Probab. Appl. 49 (2005), no. 4, 629–644. MR 2142564, DOI https://doi.org/10.1137/S0040585X97981330
- Vladimir S. Korolyuk and Nikolaos Limnios, Compound Poisson approximation with drift for stochastic additive functionals with Markov and semi-Markov switching, Recent advances in applied probability, Springer, New York, 2005, pp. 279–297. MR 2102958, DOI https://doi.org/10.1007/0-387-23394-6_13
- Vladimir S. Korolyuk and Nikolaos Limnios, Lévy approximation of increment processes with Markov switching, Stoch. Stoch. Rep. 76 (2004), no. 5, 383–394. MR 2096727, DOI https://doi.org/10.1080/10451120412331292207
- Vladimir V. Anisimov, Diffusion approximation in switching stochastic models and its applications, Exploring stochastic laws, VSP, Utrecht, 1995, pp. 13–40. MR 1713990
- Gopinath Kallianpur and Rajeeva L. Karandikar, Introduction to option pricing theory, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1718056
- X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, École d’Été de Probabilités de Saint-Flour, IV-1974, Springer, Berlin, 1975, pp. 1–96. Lecture Notes in Math., Vol. 480 (French). MR 0413238
References
- Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, 2008. MR 2378138 (2008m:60064)
- D. F. Kuznetsov, Stochastic Differential Equations: Theory and Practice of Numerical Solution, Sankt-Peterburgskii Gosudarstvennyi Politekhnicheskii Universitet, St. Petersburg, 2007. (Russian) MR 2510680 (2010h:60175)
- D. Nualart and A. Răşcanu, Differential equation driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1999. MR 1214374 (94b:60069)
- G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Ural. Gos. Univ., Sverdlovsk, 1988; English transl., Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1335454 (96e:65003)
- V. S. Korolyuk and N. Limnios, Poisson approximation of increment processes with Markov switching, Theory Prob. Appl. (2005), no. 4, 629–644. MR 2142564 (2006b:60183)
- V. S. Korolyuk and N. Limnios, Compound Poisson Approximation with Drift for Stochastic Additive Functionals with Markov and Semi-Markov Switching, Springer, New York, 2005, pp. 279–297. MR 2102958
- V. S. Korolyuk and N. Limnios, Lévy approximation of increment processes with Markov switching, Stochastics Stochastics Rep. 76 (2004), no. 5, 383–394. MR 2096727 (2005h:60101)
- V. V. Anisimov, Diffusion approximation in switching stochastic models and its application, Exploring Stochastic Laws, VSP, 1995, pp. 13–40. MR 1713990 (2001c:60142)
- G. Kallianpur and R. Karandikar, Introduction to Option Pricing Theory, Birkhäuser, Boston, 2000. MR 1718056 (2000k:91054)
- X. M. Fernique, Régularité des trajectoires de fonctions aléatoires gaussiennes, Lect. Notes in Math., vol. 480, Springer, Berlin, 1975. MR 0413238 (54:1355)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H10
Retrieve articles in all journals
with MSC (2010):
60H10
Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
myus@univ.kiev.ua
A. V. Shvaĭ
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
ShvajSS@ukr.net
Keywords:
Fractional Brownian motion,
approximate solution of stochastic differential equations
Received by editor(s):
July 8, 2009
Published electronically:
August 4, 2011
Additional Notes:
The first author is indebted to the European Commission for support in the framework of the “Marie Curie Actions” program, grant PIRSES-GA-2008-230804
The second author, Alexander Shvaĭ, died tragically on October 10, 2010, after the original Ukrainian issue had already been published
Article copyright:
© Copyright 2011
American Mathematical Society
