(Differencialnie Uravnenia i Protsesy Upravlenia)

About
History
Editorial Page
Addresses
Scope
Editorial Staff
Submission Review
For Authors
Publication Ethics
Issues
Русская версия

**Dmitriy Feliksovich Kuznetsov**

Peter the Great Saint-Petersburg Polytechnic University

Russia, 195251, Saint-Petersburg, Polytechnicheskaya st., 29

Department of Higher Mathematics

Professor, Doctor of Physico-Mathematical Sciences

In this book the problem of strong (mean-square) approximation of multiple Ito and Stratonovich stochastic integrals is systematically analyzed in the context of numerical integration of stochastic differential Ito equations. The presented monograph successfully uses the tool of multiple and iterative Fourier series, built in the space L2 and pointwise, for the strong approximation of multiple stochastic integrals and opens a new direction in researching of multiple Ito and Stratonovich stochastic integrals. We obtained a general result connected with expansion of multiple Ito stochastic integrals of any fixed multiplicity k, based on generalized multiple Fourier series converging in the space L2([t, T] x ... x [t, T]) (k-times). This result is adapted for multiple Stratonovich stochastic integrals of 1-4 multiplicity for Legendre polynomial system and system of trigonometric functions, as well as for other types of multiple stochastic integrals. The theorem on expansion of multiple Stratonovich stochastic integrals with any fixed multiplicity k, based on generalized iterative Fourier series converging pointwise is verified. We also obtained exact and approximate expressions for mean-square errors of approximation of multiple Ito stochastic integrals of multiplicity k. Significant practical material devoted to approximation of specific multiple Ito and Stratonovich stochastic integrals of 1-5 multiplicity using the system of Legendre polynomials and the system of trigonometric functions has been provided. The formulated in the book methods are compared with existing ones. We considered some weak approximations of multiple Ito stochastic integrals and proved the theorems about integration order replacement both for multiple Ito stochastic integrals and the multiple stochastic integrals according to martingales and martingale Poisson measures. Two families of analytical formulas for calculation of stochastic integrals were brought out. This book will be interesting for specialists dealing with the theory of stochastic processes, applied and computational mathematics as well as senior students and postgraduates of universities and technical institutes.

- approximation
- Fourier series
- Legendre polynomial
- mean-square convergence
- multiple Fourier-Legendre series
- Multiple Ito stochastic integral
- multiple Stratonovich stochastic integral
- multiple trigonometric
- Parseval equality
- stochastic differential Ito equation
- Stochastic Taylor expansion

- Gikhman, I. I., Skhorokhod A. V. Introduction to Theory of Stochasic Processes. (In Russian). Moscow, Nauka Publ., 1977. 660 p
- Gikhman, I. I., Skhorokhod A. V. Stochastic Differential Equations. (In Russian). Kiev, Naukova Dumka Publ., 1968. 354 p
- Gikhman, I. I., Skhorokhod A. V. Theory of Stochastic Processes. Vol. 3. (In Russian). Moscow, Nauka Publ., 1975. 469 p
- Gikhman I. I., Skhorokhod A. V. Stochastic Differential Equations and its Applications. (In Russian). Kiev, Naukova Dumka Publ., 1982 , 612 p
- Koroluk V. S., Portenko N. I., Skhorokhod A. V., Turbin A. F. Reference Book on Probability Theory and Mathematical Statistics. (In Russian). Moscow, Nauka Publ., 1985. 640 p
- Ermakov S. M., Mikhailov G. A. Course on Statistic Modeling. (In Russian). Moscow, Nauka Publ., 1976. 320 p
- Shiraev A. N. Foundations of Stochastic Finantial Mathematics. Vol. 2. (In Russian). Moscow, Fazis Publ., 1998. 544 p
- Merton R. C. Option pricing when underlying stock returns and discontinuous.
*J. Financial Economics.*3 (1976), 125-144 - Merton R. C. Continuous-Time Finance. Oxford, N. Y., Blackwell Publ., 1990. 453 p
- Hull J. Options, Futures and other Derivatives Securities. N. Y., J. Willey and Sons Publ., 1993. 368 p
- Bachelier L. Theorie de la speculation.
*Ann. Sci. Ecol. Norm. Sup*. Ser. 3. 17 (1900), 21-86 - Arato M. Linear Stochastic Systems with Constant Coefficients. A Statistical Approach. Berlin, Heidelberg, N. Y., Springer-Verlag Publ., 1982. 289 p
- Orlov A. Service of Breadth. (In Russian). Moscow, Akad. Nauk. SSSR Publ., 1958. 126 p
- Lotka A. J. Undamped oscillations derived from the law of mass action.
*J. Amer. Chem. Soc*. 42: 8 (1920), 1595-1599 - Volterra V. Mathematical Theory of Figth for Existence. (In Russian). Moscow, Nauka Publ., 1976. 286 p
- Zhabotinskiy A. M. Concentrations Self-Oscillations. (In Russian). Moscow, Nauka Publ., 1974. 178 p
- Romanovskiy Yu. M., Stepanova N. V., Chernavskiy D. S. Mathematical Biophisics. (In Russian). Moscow, Nauka Publ., 1984. 304 p
- Obuhov A. M. Description of turbulence in Lagrangian variables.
*Adv. Geophis*. (In Russian). 3 (1959), 113-115 - Wolf J. R. Neue Untersuchungen uber die Periode der Sonnenflecken und ihre Bedeutung.
*Mit. Naturforsch. Ges. Bern.*255, (1852), 249-270 - Sluzkiy E. E. On 11-years periodicity of Sun spots.
*Dokl. Akad. Nauk SSSR*. (In Russian). 4: 9, 1-2 (1935), 35-38 - Watanabe S., Ikeda N. : Stochastic Differential Equations and Diffusion Processes. (In Russian). Moscow, Nauka Publ., 1986. 445 p
- Kloeden P. E., Platen E. The Stratonovich and Ito-Taylor expansions.
*Math. Nachr.*151 (1991), 33-50 - Milstein G. N. Numerical Integration of Stochastic Differential Equations. (In Russian). Sverdlovsk, Ural University Publ., 1988. 224 p
- Kloeden P. E., Platen E. Numerical Solution of Stochastic Differential Equations. Berlin, Springer-Verlag Publ., 1992. 632 p
- Milstein G. N., Tretyakov M. V. Stochastic Numerics for Mathematical Physics. Berlin, Springer-Verlag Publ., 2004. 596 p
- Kloeden P. E., Platen E., Wright I. W. The approximation of multiple stochastic integrals.
*Stoch. Anal. Appl*. 10: 4 (1992), 431-441 - Skhorokhod A. V. Stochastc Processes with Independent Increments. (In Russian). Moscow, Nauka Publ., 1964. 280 p
- Gobson E. V. Theory of Spherical and Ellipsoidal Functions. (In Russian). Moscow, IL Publ., 1952. 476 p
- Chung K. L., Williams R. J. Introduction to Stochastic Integration. Progress in Probability and Stochastics. Vol. 4, Ed. Huber P., Rosenblatt M. Boston, Basel, Stuttgart, Birkhauser Publ., 1983. 152 p
- Averina T. A., Artemjev S. S. New family of numerical methods for solution of stochastic differential equations. (In Russian).
*Dokl. Akad. Nauk SSSR*. 288: 4 (1986), 777-780 - Kuznetsov D. F. Theorems about integration order replacement in multiple Ito stochastic integrals. (In Russian).
*VINITI*. 3607-V97 (1997), 31 p - Kuznetsov D. F. Some Problems of the Theory of Numerical Integration of Stochastic Differential Ito Equations. St. -Petersburg, Polytechnic University Publ., 1998. 203 p
- Kuznetsov D. F. Application of methods of approximation of multiple Stratonovich and Ito stochastic integrals to numerical modeling of controlled stochastic systems. (In Russian).
*Problems of Control and Informatics.*4 (1999), 91-108 - Kuznetsov D. F. Expansion of multiple Stratonovich integrals, based on multiple Fourier serieses. (In Russian).
*Zap. Nauchn. Sem. POMI Steklova V. A.*260 (1999), 164-185 - Kuznetsov D. F. On problem of numerical modeling of stochastic systems. (In Russian).
*Vestnik Molodyh Uchenyh. Prikl. Mat. Mech.*1 (1999), 20-32 - Kuznetsov D. F. Application of Legendre polynomials to mean-square approximation of solutions of stochastic differential equations. (In Russian).
*Problems of Control and Informatics*. 5 (2000), 84-104 - Kuznetsov D. F. Numerical Modeling of Stochastic Differential Equations and Stochastic Integrals. (In Russian). St. -Petersburg, Nauka Publ., 1999. 460 p
- Kuznetsov D. F. Numerical Integration of Stochastic Differential Equations. (In Russian). St. -Petersburg, State University Publ., 2001. 712 p
- Kuznetsov D. F. New representations of explicit one-step numerical methods for stochastic differential equations with jump component. (In Russian).
*Journal of Computational Mathematics and Mathematical Phisics*. 41: 6 (2001), 922-937 - Kuznetsov D. F. Multiple Stochastics Ito and Stratonovich Integrals and Multiple Fourier Serieses. (In Russian).
*Electronic Journal " Differential Equations and Control Processes".*2010, no. 3. Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book.pdf - Kuznetsov D. F. Combined method of strong approximation of multiple stochastic integrals. (In Russian).
*Problems of Control and Informatics.*4 (2002), 141-147 - Kuznetsov D. F. Numerical Integration of Stochastic Differential Equations. 2. (In Russian). St. -Petersburg, Polytechnic University Publ., 2006. 764 p
- Kuznetsov D. F. New representations of Taylor-Stratonovich expansion. (In Russian).
*Zap. Nauchn. Sem. POMI Steklova V. A.*278 (2001), 141-158 - Kuznetsov D. F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. 4th edn. (In Russian). St. -Petersburg, Polytechnic University Publ., 2010. 816 p
- Kuznetsov D. F. A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. (In Russian).
*Electronic Journal " Differential Equations and Control Processes"*. 1997, no. 1. Available at: http://www.math.spbu.ru/diffjournal/pdf/j002.pdf - Kulchitskiy O. Yu., Kuznetsov D. F. Unified Taylor-Ito expansion. (In Russian).
*Zap. Nauchn. Sem. POMI Steklova V. A.*244 (1997), 186-204 - Kulchitskiy O. Yu., Kuznetsov D. F. Approximation of multiple Ito stochastic integrals. (In Russian).
*VINITI.*1679-V94 (1994), 38 p - Dmitriy Kuznetsov. Approximation of Multiple Ito and Stratonovich Stochastic Integrals. Multiple Fourier Series Approach. Saarbrucken, AV Akademikerverlag, 2012. 344 p
- Dmitriy Kuznetsov. Numerical Integration of Ito Stochastic Differential Equations. With MatLab Programms. (In Russian). Saarbrucken, AV Akademikerverlag, 2012. 692 p
- Dynkin E. B. Markov's Processes. (In Russian). Moscow, Fizmatgiz Publ., 1963. 860 p
- Kamke E. Reference Book on Ordinary Differential Equations. Vol. 1. (In Russian). Moscow, Nauka Publ., 1971. 576 p
- Stratonovich R. L. Conditional Markov's Processes and its Applications to the Theory of Optimal Control. (In Russian). Moscow, State University Publ., 1966. 320 p