ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Stochastic Differential Equations: Theory and Practic of Numerical Solution. With MATLAB Programs

Author(s):

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

sde_kuznetsov@inbox.ru

Abstract:

This is the sixth (revised and expanded) edition of the book "Stochastic differential equations: theory and practice of numerical solution".
The monograph is devoted to the problem of the numerical integration of stochastic differential equations (SDE). The case of the Ito SDE is systematically analised and the case of SDE with jump component is considered as well. In the book the effective approach to the numerical integration of the Ito SDE, which is based on the Taylor-Ito and the Taylor-Stratonovich expansions, is systematically analised.
One of the aims of this monograph is the development and application of the Fourier method to the numerical solution of the Ito SDE. The Fourier series are widely used in various fields of applied mathematics and physics. However, the method of the Fourier series as applied to the numerical solution of the Ito SDE has not been adequately studied.
This monograph partially fills this gap. The book consists of 17 chapters. In chapter 1 we present additional material which may be used while reading this book. In chapter 2 the mathematical models of dynamical systems of different physical nature under the influence of random disturbances on the base of SDE are considered. Some mathematical problems for SDE are described as well. Chapter 3 is devoted to some properties and formulas for stochastic integrals. In chapter 4 we discuss the stochastic Taylor expansions.
We consider the classical Taylor-Ito and Taylor-Stratonovich expansions and 4 new, so called unified the Taylor-Ito and the Taylor-Stratonovich expansions.
Chapters 5 and 6 are devoted to strong (mean-square) approximations of collections of the iterative Ito and Stratonovich stochastic integrals from the Taylor-Ito and Taylor-Stratonovich expansions. In chapters 7-9 we construct strong numerical methods, and in chapter 10 we consider weak numerical methods for the Ito SDE.
Chapter 11 is devoted to the numerical integration of linear stationary systems of the Ito SDE. In chapter 12 we consider the theory of numerical integration of SDE with jump component. In chapter 13 we provide the library of MATLAB programs for the numerical integration of linear stationary systems of the Ito SDE. In chapters 14-16 the application of the numerical methods constructed in the monograph to the modeling selected trajectories of solutions of the non-linear Ito SDE systems (chapter 14), to numerical solution of mathematical problems by strong (chapter 15) and weak (chapter 16) numerical methods is demonstrated.
For the first time the numerical modeling the iterative Ito and Stratonovich stochastic integrals is realised with the usage of the Legendre polynomial system. Chapter 17 contains full texts of MATLAB programs, implementing the numerical experiments along the text of the whole book.

Keywords

References:

  1. Averina T. A., Artem’ev S. S. New family of numerical methods for solving of stochastic differential equations. Dokl. Ak. Nauk SSSR, 288: 4 (1986), 777-780. (In Russ. )
  2. Averina T. A., Artem’ev S. S. Numerical solution of stochastic differential equations with growing variance. Sibirskiy zh. vychisl. matem. 8: 1 (2005), 1-10 (In Russ. )
  3. Averina T. A., Karachanskaya E. V., Rybakov K. A. [Modeling and analysis of linear invariant stochastic systems]. Differencial’nie uravnenia i processy upravlenia, 2018, no. 1, 54-76 (In Russ. ). Available at: http://www.math.spbu.ru/diffjournal/pdf/rybakov7.pdf
  4. Arato M., Kolmogorov A. N., Sinay Ja. G. About estimates of parameters of complex stationary Gaussian Markov process. Dokl. Akad. Nauk SSSR. 146: 4 (1962), 747-750. (In Russ. )
  5. Arsenin V. Ya. Metody matematicheskoi fiziki i special’nye funkcii [Methods of mathematical physics and special functions]. Moscow, Nauka Publ., 1974. 431 p
  6. Arsen’ev D. G., Kulchitsky O. Yu. Optimization of algorithms of numerical integration of stiff linear systems of differential equations with constant coefficients. VINITI, 732-V86 (1986), 32 p. (In Russ. )
  7. Artem’ev S. S. Stability in the mean-square of numerical methods for solving of stochastic differential equations. Dokl. Akad. Nauk SSSR. . 333: 4 (1993), 421-424 (In Russ. )
  8. Artem’ev S. S., Shkurko I. O. [Numerical solution of linear systems of stochastic differential equations] Trudy VII Vsesoyuzn. Soveschaniya “Metody Monte-Karlo v vychislitel’noi matematike i matematicheskoi fizike” [Proc. Conf “Monte-Carlo methods in computational mathematics and mathematical physics”], Novosibirsk, 1985. pp. 144-146. (In Russ. )
  9. Artem’ev S. S., Yakunin M. A. Matematicheskoye i statisticheskoye modelirovanie v finansah [Mathematical and statistical modeling in finances], Novosibirsk, IVMMG SO RAN Publ., 2008. 174 p
  10. Atalla M. A. [Finite-difference approximations for stochastic differential equations]. Sb. Nauchn. Trudov Inst. Mat. Ak. Nauk. Ukr. SSR “Veroyatnostnye metody issledovaniya system s beskonechnym chislom stepenei svobody” [Proc. of the Inst. of Math. of the Akad. of Sc. of Ukr. SSR “Probabilistic methods for the study of systems with an infinite number of degrees of freedom”], Kiev, 1986, pp. 11-16. (In Russ. )
  11. Auslender E. I., Milstein G. N. Asymptotic expansion of Lyapunov exponent for linear stochastic systems with small noises. Prikl. Mat. i Mekh. 46: 3 (1982), 358-365. (In Russ. )
  12. Barkin A. I., Zelentsovsky A. L., Pakshin P. V. Absol’utnaya ustoichivost’ determinirovannyh i stokhasticheskih system upravleniya [Absolute stability of deterministic and stochastic control systems]. Moscow, MAI Publ., 1992. 303 p
  13. Bahvalov N. S. Chislennye metody [Numerical methods]. Moscow, Fizmatgiz Publ., 1973. 631 p
  14. Belousov B. P. [Periodically acting reaction and its mechanism] Sb. referatov po radiacionnoi medicine [A collection of abstracts on radiation medicine], Moscow, Medgiz Publ., 1959, pp. 145-148 (In Russ. )
  15. Volterra V. Matematicheskaya teoriya bor’by za suschestvovanie [Mathematical theory of struggle for existence], Moscow, Nauka Publ., 1976. 286 p
  16. Girsanov I, V. About the conversion of one class of stochastic processes with absolutely continuous substitution of measure. Teor. Veroyatn. i prim. . 5: 3 (1960), 314-330. (In Russ. )
  17. Gihman I. I., Scorokhod A. V. Stokhasticheskie differencial’nye uravnenia [Stochastic differential equations]. Kiev, Naukova Dumka Publ., 1968. 354 p
  18. Gihman I. I., Scorokhod A. V. Teoria sluchainyh processov. T. 3 [The theory of stochastic processes. V. 3]. Moscow, Nauka Publ., 1975. 469 p
  19. Gihman I. I., Scorokhod A. V. Vvedenie v teoriu sluchainyh processov [Introduction to the theory of stochastic processes]. Moscow, Nauka Publ., 1977. 660 p
  20. Gihman I. I., Scorokhod A. V. Stokhasticheskie differencial’nye uravnenia i ih prilozhenia [Stochastic differential equations and its applications]. Kiev, Naukova Dumka Publ., 1982. 612 p
  21. Gladyshev S. A., Milstein G. N. Method Runge-Kutta for calculating of Wiener integrals of exponential type. Zh. Vychisl. Mat. i Mat. Phiz. . 24 (1985), 1136-1149. (In Russ. )
  22. Djakonov V. P. Spravochnic po primeneniu sistemy PC MatLab [Handbook on application of PC MatLab]. Moscow, Nauka Publ., 1993. 111 p
  23. Djakonov V, P. MATLAB 6. 5 SP1/7. 0 + Simulink 5/6. Osnovy primeneniya. [MATLAB 6. 5 SP1/7. 0 + Simulink 5/6. The Basics of application]. Moscow, SOLON-press Publ., 2005. 800 p
  24. Dzagnidze Z. A., Chitashvili R. Ya. [Approximate integration of stochastic differential equations]. Sb. Nauchn. Trudov Inst. Prikl. Mat. Tbil. Gos. Univ. ”Trudy IV” [Proc. of the Inst. of Appl. Math. of the Tbil. St. Univ. ], 1975, pp. 267-279. (In Russ. )
  25. Dynkin E. B. Markovskie processy [Markov processes] Moscow, Nauka Publ., 1963. 860 p
  26. Ermakov S. M., Mikhailov G. A. Kurs statisticheskogo modelirovania [The course of statistical modeling]. Moscow, Nauka Publ., 1976. 320 p
  27. Zhabotinsky A. M. Koncentracionnye avtokolebaniya [Concentration self-oscillations], Moscow, Nauka Publ., 1974. 178 p
  28. Zhizhiashvili L. V. Soprjazhennye funkcii i trigonometricheskije r'jady [Conjugate functions and trigonometric series]. Tbilisi, Tbil. Univ. Publ., 1969. 271 p
  29. Il’in V. A., Poznyak E. G. Osnovy matematicheskogo analiza. Chast’ 2 [Foundations of mathematical analysis. Part II]. Moscow, Nauka Publ., 1973. 448 p
  30. Ito K. Veroyatnostnye processy. Vyp. 2 [Probubilistic processes. Vol. 2], Moscow, IL Publ., 1963. 135 p
  31. Kamke E. Spravochnik po obyknovennym differencial’nym uravneniam. T. 1 [Handbook on ordinary differential equations. Vol. 1]. Moscow, Nauka Publ., 1971. 576 p
  32. Kozin F. Introduction to stability of stochastic systems. Avtomatika. 5 (1969), 95-112. (In Russ. )
  33. Korenevsky M. L. [About optimization of one method of approximate calculation of matrix exponent]. Trudy Mezhdunarodnoi Konferencii “Sredstva matematicheskogo modelirovania” [Proc. Int. Conf. “Tools of Mathematical Modeling”], St. -Petersburg, 1998, pp. 125-134. (In Russ. )
  34. Koroluk V. S., Portenko N. I., Skorokhod A. V., Turbin A. F. Spravochnik po teorii veroyatnostei i matematicheskoi statistike [Handbook on the probability theory and mathematical statistics]. Moscow, Nauka Publ., 1985. 640 p
  35. Kuznetsov D. F. Finite-difference approximation of Taylor-Ito expansion and finite-difference methods for numerical integration of Ito stochastic differential equations. VINITI, 3509-V96 (1996), 24 p. (In Russ. )
  36. Kuznetsov D. F. Finite-difference method with local mean-square error of order 3 for numerical integration of Ito stochastic differential equations. VINITI, 3510-V96 (1996), 27 p. (In Russ. )
  37. Kuznetsov D. F. Theoretical substantiation of the method of expansion and approximation of repeated Stratonovich stochastic integrals based on multiple Fourier series on trigonometric and spherical functions. VINITI, 3608-V97 (1997), 27 p. (In Russ. )
  38. Kuznetsov D. F. Theorems about integration order replacement in multiple stochastic integrals. VINITI, 3607-V97 (1997), 31 p. (In Russ. )
  39. Kuznetsov D. F. [A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems]. Differencial’nie uravnenia i processy upravlenia, 1997, no. 1, 18-77 (In Russ. ) Available at: http://www.math.spbu.ru/diffjournal/pdf/j002.pdf
  40. Kuznetsov D. F. [Some problems of the theory of numerical solution of Ito stochastic differential equations]. Differencial’nie uravnenia i processy upravlenia, 1998, no. 1, 66-367. (In Russ. ). DOI: 10. 18720/SPBPU/2/z17-6. Available at: http://www.math.spbu.ru/diffjournal/pdf/j011.pdf
  41. Kuznetsov D. F. [Analitic Formulae for Computing Stochastic Integrals]. Differencial’nie uravnenia i processy upravlenia, 1998, no. 4, 18-28 (In Russ. ). Available at: http://www.math.spbu.ru/diffjournal/pdf/j025.pdf
  42. Kuznetsov D. F. [A method for the expansion and approximation of iterated Stratonovich stochastic integrals that is based on multiple Fourier series in complete orthonormalized systems of functions, and its application to the numerical solution of Ito stochastic differential equations]. Trudy Mezhdunarodnoi Konferencii “Sredstva matematicheskogo modelirovania” [Proc. Int. Conf. “Tools of Mathematical Modeling”], St. -Petersburg, 1998, pp. 135-160. (In Russ. )
  43. Kuznetsov D. F. [Application of different full orthonormal systems of functions for numerical solution of Ito stochastic differential equations] Trudi Mezhdunarodnoi Konferencii " Differencial’nye Uravnenia i ih Primenenia" [Proc. Int. Conf. “Differential Equations and its Applications”], St. -Petersburg, 1998, pp. 128-129 (In Russ. )
  44. Kuznetsov D. F. Application of approximation methods of iterated Stratonovich and Ito stochastic integrals to numerical simulation of controlled stochastic systems. ProblemyUpravleniya i Informatiki 4 (1999), 91-108 (In Russ. )
  45. Kuznetsov D. F. An expansion of multiple Stratonovich stochastic integrals, based on multiple Fourier expansion. Zap. Nauchn. Sem. POMI im. V. A. Steklova. 260 (1999), 164-185. (In Russ. )
  46. Kuznetsov D. F. On problem of numerical modeling of stochastic systems. Vestnik Molodykh Uchenyh. Serya: Prikl. Mat. i Mekh. 1 (1999), 20-32. (In Russ. )
  47. Kuznetsov D. F. Chislennoye modelirovanie stokhasticheskih differencial’nyh uravnenii i stokhasticheskih integralov [Numerical modeling of stochastic differential equations and stochastic integrals]. St. -Petersburg, Nauka Publ., 1999. 460 p
  48. Kuznetsov D. F. Mean-square approximation of solutions of stochastic differential equations using Legendre’s polynomials. Problemy Upravleniya i Informatiki 5 (2000), 84-104 (In Russ. )
  49. Kuznetsov D. F Weak numerical method of order 4. 0 for stochastic differential Ito equations. Vestnik Molodykh Uchenyh. Serya: Prikl. Mat. i Mekh. . 4 (2000), 47-52. (In Russ. )
  50. Kuznetsov D. F Chislennoye integrirovanie stokhasticheskih differencial’nyh uravnenii [Numerical Integration of Stochastic Differential Equations], St. -Petersburg, State Univ. Publ.,. 2001. 712 p
  51. Kuznetsov D. F. New representations of explicit one-step numerical methods for jump-diffusion stochastic differential equations. Zh. Vychisl. Mat. i Mat. Phiz. 41: 6 (2001), 922-937. (In Russ. )
  52. Kuznetsov D. F. New representations of the Taylor-Stratonovich expansion. Zap. Nauchn. Sem. POMI im. V. A. Steklova. 278 (2001), 141-158. (In Russ. )
  53. Kuznetsov D. F. Finite-difference strong numerical methods of order 1. 5 and 2. 0 for stochastic differential Ito equations with nonadditive multidimensional noise. Problemy Upravleniya i Informatiki 4 (2001), 59-73. (In Russ. )
  54. Kuznetsov D. F. Combined method of strong approximation of multiple stochastic integrals. Problemy Upravleniya i Informatiki 4 (2002), 141-147. (In Russ. )
  55. Kuznetsov D. F. The three-step strong numerical methods of the orders of accuracy 1. 0 and 1. 5 for Ito stochastic differential equations Problemy Upravleniya i Informatiki. 6 (2002), 104-119. (In Russ. )
  56. Kuznetsov D. F. Chislennoye integrirovanie stokhasticheskih differencial’nyh uravnenii. 2. [Numerical Integration of Stochastic Differential Equations. 2. ], St. -Petersburg, Polytechnic Univ. Publ., 2006. 764 p. DOI: 10. 18720/SPBPU/2/s17-227
  57. Kuznetsov D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. . S programmoi dl’ja PC v sisteme MATLAB 7. 0. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 program. ], St. -Petersburg, Polytechnic Univ. Publ., 2007, 778 p. DOI: 10. 18720/SPBPU/2/s17-228
  58. Kuznetsov D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. . S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 2-e. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 programs . 2-th Ed. ], St. -Petersburg, Polytechnic Univ. Publ., 2007, xxxii+770 p. DOI: 10. 18720/SPBPU/2/s17-229
  59. Kuznetsov D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 3-e. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 programs. 3-th Ed. ], St. -Petersburg, Polytechnic Univ. Publ., 2009, xxxiv+768 p. DOI: 10. 18720/SPBPU/2/s17-230
  60. Kuznetsov D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. . S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 4-e. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. . With MATLAB 7. 0 programs . 4-th Ed. ], St. -Petersburg, Polytechnic Univ. Publ., 2010, xxx+786 p. DOI: 10. 18720/SPBPU/2/s17-231
  61. Kuznetsov D. F. [Multiple Stochastic Ito and Stratonovich Integrals and Multiple Fourier Serieses]. Differencial’nie uravnenia i processy upravlenia, 2010, no. 3, A. 1-A. 257 (In Russ. ). DOI: 10. 18720/SPBPU/2/z17-7. Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book.pdf
  62. Kuznetsov D. F. [Stochastic differential equations: theory and practice of numerical solution. With Matlab programs. 5-th Ed. ]. Differencial’nie uravnenia i processy upravlenia, 2017, no. 2, A. 1-A. 1000 (In Russ. ). DOI: 10. 18720/SPBPU/2/z17-4. Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book3.pdf
  63. Kuznetsov D. F. [Expansion of Multiple Stratonovich Stochastic Integrals of Second Multiplicity, Based on Double Fourier-Legendre Series Summarized by Prinsheim Method]. Differencial’nie uravnenia i processy upravlenia, 2018, no. 1, 1-34 (In Russ. ). Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_4.pdf
  64. Kuznetsov D. F. On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1. 5 and 2. 0 orders of strong convergence. Avtomat. i Telemekh. . 7 (2018), 80-98 . (In Russ. )
  65. Kulchitsky O. Yu., Kuznetsov D. F. Expansion of Ito processes into Taylor-Ito series at the neighborhood of the fixed time moment. VINITI, 2637-V93 (1993), 26 p. (In Russ. )
  66. Kulchitsky O. Yu., Kuznetsov D. F. Approximation of multiple Ito stochastic integrals. VINITI, 1678-V94 (1994), 42 p. (In Russ. )
  67. Kulchitsky O. Yu., Kuznetsov D. F. [The unified Taylor-Ito expansion. Унифицированное разложение Тейлора-Ито]. Differencial’nie uravnenia i processy upravlenia, 1997, no. 1, 1-17. Available at: http://www.math.spbu.ru/diffjournal/pdf/j001.pdf
  68. Kulchitsky O. Yu., Kuznetsov D. F. The Unified Taylor-Ito expansion. Zap. Nauchn. Sem. POMI im. V. A. Steklova. 244 (1997), 186-204. (In Russ. )
  69. Kulchitsky O. Yu., Kuznetsov D. F. [Numerical simulation of stochastic systems of linear stationary differential equations]. Differencial’nie uravnenia i processy upravlenia, 1998, no. 1, 41-65 (In Russ. ). Available at: http://www.math.spbu.ru/diffjournal/pdf/j010.pdf
  70. Kulchitsky O. Yu., Kuznetsov D. F. Numerical methods of modeling control systems described by stochastic differential equations. Problemy Upravleniya i Informatiki. 2 (1998), 57-72 (In Russ. )
  71. Ladyzhenskaya O. A., Solonnikov V. A., Ural’ceva N. N. Lineinye i kvazilineinye uravnenia parabolicheskogo tipa [Linear and quasilinear equations of parabolic type]. Moscow, Nauka Publ., 1967. 736 p
  72. Liptser R. Sh., Shirjaev A. N. Statistika sluchainyh processov: nelineinaya fil’tracia i smezhnye voprosy [Statistics of stochastic processes: nonlinear filtering and related problems]. Moscow, Nauka Publ., 1974. 696 p
  73. Milstein G. N. Approximate integration of stochastic differential equations. Teor. Veroyatn. i Primen. 19 (1974), 557-562. (In Russ. )
  74. Milstein G. N. A Method of second-order accuracy of integration of stochastic differential equations. Teor. Veroyatn. i Primen 23 (1978), 396-401. (In Russ. )
  75. Milstein G. N. Weak approximation of solutions of systems of stochastic differential equations. Teor. Veroyatn. i Primen. 30 (1985), 750-766. (In Russ. )
  76. Milstein G. N. Chislennoye integrirovaniye stokhasticheskih differencial’nyh uravnenii [Numerical integration of stochastic differential equations], Sverdlovsk, Ural. Univ. Publ., 1988. 225 с
  77. Milstein G. N. The Solution of the first boundary problem for equations of parabolic type by integrating of stochastic differential equations. Teor. Veroyatn. i Primen. 40 (1995), 657-665. (In Russ. )
  78. Nasyrov F. S. Lokal’nye vremena, simmetrichnyje integraly i stochasticheskiy analiz [Local times, symmetric integrals and stochastic analysis]. Moscow, Fizmatlit Publ., 2011. 212 p
  79. Neymark Yu. I., Landa P. S. Stokhasticheskie i haoticheskie kolebaniya [Stochastic and chaotic oscillations], Moscow, Nauka Publ., 1987. 424 p
  80. Nikitin N. N., Razevig V. D. Methods of digital modeling of stochastic differential equations and estimate of their errors. Zh. Vychisl. Mat. i Mat. Phiz. 18: 1 (1978), 106-117. (In Russ. )
  81. Orlov A. Sluzhba Shiroty [Service of Latitude], Moscow, Akad. Nauk SSSR Publ., 1958. 126 p
  82. Pervozvansky A. A. Rynok: raschet i risk [Market: calculation and risk], Moscow, INFRA Publ., 1994. 210 p
  83. Pontrjagin L. S. Obyknovennye differencial’nye uravnenia. Izd. 5-e. [Ordinary differential equations. 5-th Ed. ]. Moscow, Nauka Publ., 1982. 331 p
  84. Pugachev V. S., Sinitsin I. N. Stokhasticheskie differencial’nye sistemy: analiz i fil’traciya. [Stochastic differential systems: analysis and filtration], Moscow, Nauka Publ., 1985. 559 p
  85. Razevig V. D. Digital modeling of multidimensional dynamic systems under stochastic disturbances. Avtomat. i Telemekh. . 4 (1980), 177-186. (In Russ. )
  86. Rozanov Yu. A. Stacionarnye sluchainye processy [Stationary stochastic processes]. Moscow, Fiznatgiz Publ., 1963. 284 p
  87. Romanovsky Yu. M., Stepanova N. V., Chernavsky D. S. Matematicheskaya biofizika [Mathematical biophysics], Moscow, Nauka Publ., 1984. 304 p
  88. Ryzhik I. M., Gradstein I. S. Tablicy integralov, sum, ryadov i proizvedenii. Izd. 3-e [Tables of integrals, sums, series, and products. 3-rd Ed. ]. Moscow-Leningrad, State Publ. House of Techn. and. Theor. Lit, 1951. 464 p
  89. Samarsky A. A., Galaktionov V. A., Kurd’umov S. P. Rezhimy s obostreniem v zadachah dl’a kvazilineinyh parabolicheskih uravnenii. [Regimes with aggravation in the problems for quasilinear parabolic equations], Moscow, Nauka Publ,, 1987. 476 p
  90. Samarsky A. A. Teoriya raznostnyh skhem. Izd. 3-e. [Theory of difference schemes. 3rd Ed. ], Moscow, Nauka Publ., 1989. 614 p
  91. Scorokhod A. V. Sluchainye processy s nezavisimymi proraschenyami [Stochastic processes with independent increments]. Moscow, Nauka Publ., 1964. 280 p
  92. Slucky E. E. About 11-year periodicity of sunspots, Dokl. Akad. Nauk SSSR, 4: 9, 1-2 (1935), 35-38. (In Russ. )
  93. Starchenko T. K. [About conditions of convergence of double Fourier-Legendre series]. Trudy inst. matematiki NAN Belarusi. " Analiticheskije metody analiza i differencial'nyh uravnenii [Proc. of the Mathematical inst. of NAS of Belarus. " Analitical methods of analysis and differential equations]. Minsk, 2005, no. 5, pp. 124-126 (in Russ. )
  94. Stratonovich R. L. Izbrannye voprosy teorii fluctuacii v radiotekhnike [Selected questions of the theory of fluctuacions in radio engineering]. Moscow, Soviet Radio Publ., 1961. 556 p
  95. Stratonovich R. L. Uslovnye markpvskie processy i ih primenenie k teorii optimal’nogo upravlenia [Conditional Markov’s processes and its application to the theory of optimal control]. Moscow, Moscow St. Univ. Publ, 1966. 320 p
  96. Stratonovich R. L., Polyakova M. S. Elementy molekulyarnoi fiziki, termodinamiki i statisticheskoi fiziki [Elements of molecular physics, thermodynamics and statistical physics]. Moscow, Moscow St. Univ. Publ, 1981. 176 p
  97. Suetin P. K. Klassicheskije ortogonal'nye mnogochleny. Izd 3-e. [Classical orthogonal polynomials. 3rd Ed. ]. Moskow, Fizmatlit, 2005. 480 p
  98. Tikhonov A. N., Samarskii A. A. Uravnenia matematicheskoi fiziki. Izd. 5-e [Equations of mathematical physics. 5-th Ed]. Moscow, Nauka Publ., 1977. 735 p
  99. Tolstov G. P. Rjady Fur’e [Fourier series]. Moscow-Leningrad, State Publ. House of Techn. and. Theor. Lit, 1951. 396 p
  100. Has’minsky R. Z. Ustoichivost’ system differencial’nyh uravnenii pri sluchainyh vozmuscheniyah ih parametrov [Stability of systems of differential equations under random disturbances of their parameters], Moscow, Nauka Publ., 1969. 365 p
  101. Shirjaev A. N. Veroyatnost’ [Probability]. Moscow , Nauka Publ., 1989. 640 p
  102. Shirjaev A. N. Osnovy stokhasticheskoi finansovoi matematiki. T. 2 [Foundations of the stochastic financial mathematics. V. 2]. Moscow, Fazis Publ., 1998. 544 p
  103. Allen E. Modeling with Ito stochastic differential equations. Dordrecht, Springer Publ., 2007. 240 p
  104. Allen E. Approximation of triple stochastic integrals through region subdivision. Communications in Applied Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  105. Arato M. Linear stochastic systems with constant coefficients. A statistical approach. Berlin, Heidelberg, N. Y., Springer-Verlag Publ., 1982. 289 p
  106. Arnold L. Stochastic differential equations: Theory and applications. N. Y., Wiley Publ., 1974. 228 p
  107. Arnold L., Kloeden P. E. Explicit formulae for the Lyapunov exponents and rotation number of two-dimensional systems with telegraphic noise. SIAM J. Appl. Math. 49 (1989), 1242-1274
  108. Bachelier L. Theorie de la speculation. Ann. Sci. Ecol. Norm. Sup. Ser. 3. 17 (1900), 21-86
  109. Bally V., Talay D. The Euler scheme for stochastic differential equations: Error analysis with Malliavin calculus. Math. Comput. Simulation. 38 (1995), 35-4l
  110. Bally V., Talay D. The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function. Probab. Theory Related Fields. 104: 1 (1996), 43-60
  111. Bally V., Talay D. The law of the Euler scheme for stochastic differential equations II. Convergence rate of the density. Monte Carlo Methods Appl. 2: 2 (1996) 93-128
  112. Bjork T., Kabanov Yu., Runggaldier W. Bond market structure in the presence of marked point processes. Math. Finance. 7: 2 (1997), 211-239
  113. Blumenthal R. M., Getoor R. K., McKean H. P. Markov processes with identical hitting distributions. Illinois J. Math. 6 (1962), 402-421
  114. Boyce W. E. Approximate solution of random ordinary differential equations. Adv. in Appl. Probab. 10 (1978), 172-184
  115. Burrage K., Platen E. Runge-Kutta methods for stochastic differential equations. Ann. Numer. Math. 1: 1-4 (1994), 63-78
  116. Chang C. C. Numerical solution of stochastic differential equations with constant diffusion coefficients. Math. Comput. 49 (1987), 523-542
  117. 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
  118. Claudine L, Rosler A. Iterated stochastic integrals in infinite dimensions - approximation and error estimates. arXiv:1709. 06961 [math. PR], 2017, 22 p
  119. Clements D. J., Anderson B. D. O. Well behaved Ito equations with simulations that always misbehave. IEEE Trans. Automat. Control. AC-18 (1973), 676-677
  120. Doob J. L. Semimartingales and subharmonic functions. Trans. Amer. Math. Soc. 77 (1954), 86-121
  121. Einstein A. Investigations on the theory of the Brownien movement. N. Y., Dover, 1956. 122 p
  122. Feng J. F. Numerical solution of stochastic differential equations. Chinese J. Numer. Math. Appl. 12 (1990), 28-41
  123. Feng J. F., Lei G. Y., Qian M. P. Second order methods for solving stochastic differential equations. J. Comput. Math. 10: 4 (1992), 376-387
  124. Friedman A. Partial differential equations of parabolic type. Englewood Cliffs, Prentice-Hall Publ., 1964. 347 p
  125. Gantmacher F. R. The theory of matrices. New York, Chelsea Publ., 1959. Vol. 1: 374 p., Vol. 2: 277 p
  126. Gard T. C. Introduction to stochastic differential equations. N. Y., Marcel Dekker Publ., 1988. 324 p
  127. Gilsing H, Shardlow T. SDELab: A package for solving stochastic differential equations in MATLAB. J. Comp. Appl. Math. 205: 2 (2007), 1002-1018
  128. Greenside H. S., Helfand E. Numerical integration of stochastic differential equations. II. Bell System Tech. J. 60 (1981), 1927-1940
  129. Han X, Kloeden P. E. Random ordinary differential equations and their numerical solution. Singapore: Springer Publ. 2017, 250 p
  130. Hardy G. H., Rogosinski W. W. Fourier series. N. Y., Dover Publ., 1999. 112 p
  131. Haworth D. C., Pope S. B. A second-order Monte-Carlo method for the solution of the Ito stochastic differential equation. Stoch. Anal. Appl. 4 (1986), 151-186
  132. Henrici P. Discrete variable methods in ordinary differential equations. N. Y., Wiley Publ., 1962. 407 p
  133. Hernandez D. B., Spigler R. Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise. BIT. 33 (1993), 654-669
  134. Higham D. J., Mao X., Stuart A. M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40 (2002), 1041-1063
  135. Hobson E. W. The theory of spherical and ellipsoidal harmonics. Cambridge, Cambridge Univ. Press, 1931. 502 p
  136. Hofmann N., Platen E. Stability of weak numerical schemes for stochastic differential equations. Comput. Math. Appl. 28: 10-12 (1994), 45-57
  137. Hofmann N., Platen E. Stability of superimplisit numerical methods for stochastic differential equations. Fields Inst. Communications. 9 (1996), 93-104
  138. Hull J., White A. The pricing of options as assets with stochastic volatilities. J. Finance. 42 (1987), 281-300
  139. Hull J. Options, futures and other derivatives securities. N. Y., J. Willey and Sons Publ., 1993. 368 p
  140. Ikeda N., Watanabe S,. Stochastic differential equations and diffusion processes. Amsterdam, Oxford, N. Y., North Holland Publ. Co., 1981. 480 p
  141. Janssen R. Difference-methods for stochastic differential equations with discontinuous coefficients. Stochastics. 13 (1984), 199-212
  142. Janssen R. Discretization of the Wiener process in difference methods for stochastic differential equations. Stochastic Process. Appl. 18 (1984), 361-369
  143. Kac M. On distribution of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949), 1-13
  144. Kac M. On some connections between probability theory and differential and integral equations. Proc. Second Berkley Symp. Math. Stat. Probab. 1 (1951), 189-215
  145. Kamrani M., Jamshidi N. Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation. Numerical Algorithms. (2017), 1-18
  146. Klauder J. R., Petersen W. P. Numerical integration of multiplicative-noise stochastic differential equations. SIAM J. Numer. Anal. 22 (1985), 1153-1166
  147. Kloeden P. E., Platen E. The Stratonovich and Ito-Taylor expansions. Math. Nachr. 151 (1991), 33-50
  148. Kloeden P. E., Platen E. Numerical solution of stochastic differential equations. Berlin, Springer-Verlag Publ., 1992. 632 p
  149. Kloeden P. E., Platen E., Wright I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10: 4 (1992), 431-441
  150. Kloeden P. E., Platen E. Higher-order implicit strong numerical schemes for stochastic differential equations. J. Statist. Phisics. 66 (1992), 283-314
  151. Kloeden P. E., Platen E., Schurz H. Numerical solution of SDE through computer experiments. Berlin, Springer-Verlag Publ., 1994. 292 p
  152. Kloeden P. E., Platen E., Hofmann N. Extrapolation methods for the weak approximation of Ito diffusions. SIAM J. Numer. Anal. 32 (1995), 1519-1534
  153. Kloeden P. E., Platen E., Schurz H., Sorensen M. On effects of discretization on estimators of drift parameters for diffusion processes. J. Appl. Probab. 33 (1996), 1061-1076
  154. Kushner H. J. Stochastic stability and control. N. Y., London, Academic Press, 1967. 162 p
  155. Kushner H. J. Probubility methods for approximations in stochastic control and for elliptic equations. N. Y., San Francisco, London, Academic Press, 1977. 242 p
  156. Kuznetsov D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach. (In English). St. -Petersburg, Polytechnical University Publishing House, 2011, 250 p. DOI: 10. 18720/SPBPU/2/s17-232
  157. Kuznetsov D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach. 2nd Ed. (In English). St. -Petersburg, Polytechnical University Publishing House, 2011, 284 p. DOI: 10. 18720/SPBPU/2/s17-233
  158. Kuznetsov D. F. Multiple Ito and Stratonovich stochastic integrals: approximations, properties, formulas. (In English). St. -Petersburg, Polytechnical University Publishing House, 2013, 382 p. DOI: 10. 18720/SPBPU/2/s17-234
  159. Kuznetsov D. F. Multiple Ito and Stratonovich stochastic integrals: Fourier-Legendre and trigonometric expansions, approximations, formulas. Electronic Journal “Differential Equations and Control Processes”. 2017, no. 1, A. 1-A. 385 (In English). DOI: 10. 18720/SPBPU/2/z17-3. Available at: http://www.math.spbu.ru/diffjournal/pdf/kuznetsov_book2.pdf
  160. Kuznetsov D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals. Intern. Conf. on Mathematical Modeling in Applied Sciences. Abstracts Book, S. -Petersburg, Polytechnic University Publishing House, 2017, pp. 141-142
  161. Kuznetsov D. F. Application of the Fourier method to the mean-square approximation of multiple Ito and Stratonovich stochastic integrals. (In English). arXiv:1712. 08991 [math. PR]. 2017, 15 p. Available at: https://arxiv.org/abs/1712.08991
  162. Kuznetsov D. F. Expansions of multiple Stratonovich stochastic integrals, based on generalized multiple Fourier series. (In English). arXiv:1712. 09516 [math. PR]. 2017, 25 p. Available at: https://arxiv.org/abs/1712.09516
  163. Kuznetsov D. F. Expansion of multiple Ito stochastic integrals of arbitrary multiplicity, based on generalized multiple Fourier series, converging in the mean. (In English). arXiv:1712. 09746 [math. PR]. 2017, 22 p. Available at: https://arxiv.org/abs/1712.09746
  164. Kuznetsov D. F. Mean-square approximation of multiple Ito and Stratonovich stochastic integrals from the Taylor-Ito and Taylor-Stratonovich expansions, using Legendre polynomials. (In English). arXiv:1801. 00231 [math. PR]. 2017, 26 p. Available at: https://arxiv.org/abs/1801.00231
  165. Kuznetsov D. F. Expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity, based on generalized repeated Fourier series, converging pointwise. (In English). arXiv:1801. 00784 [math. PR]. 2018, 23 p. Available at: https://arxiv.org/abs/1801.00784
  166. Kuznetsov D. F. Exact calculation of mean-square error of approximation of multiple Ito stochastic integrals for the method, based on the multiple Fourier series. (In English). arXiv:1801. 01079 [math. PR]. 2018, 19 p. Available at: https://arxiv.org/abs/1801.01079
  167. Kuznetsov D. F. Expansion of triple Stratonovich stochastic integrals, based on generalized multiple Fourier series, converging in the mean: general case of series summation. (In English). arXiv:1801. 01564 [math. PR]. 2018, 26 p. Available at: https://arxiv.org/abs/1801.01564
  168. Kuznetsov D. F. Expansion of multiple Stratonovich stochastic integrals of multiplicity 2, based on double Fourier-Legendre series, summarized by Prinsheim method. (In Russ. ). arXiv:1801. 01962 [math. PR]. 2018, 21 p. Available at: https://arxiv.org/abs/1801.01962
  169. Kuznetsov D. F. The hypothesis about expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity. (In English). arXiv:1801. 03195 [math. PR]. 2018, 14 p. Available at: https://arxiv.org/abs/1801.03195
  170. Kuznetsov D. F. Direct combined approach for expansion of multiple Stratonovich stochastic integrals of multiplicities 2 - 4, based on generalized multiple Fourier series. (In English). arXiv:1801. 05654 [math. PR]. 2018, 22 p. Available at: https://arxiv.org/abs/1801.05654
  171. Kuznetsov D. F. Application of the direct combined approach to expansion of double Stratonovich stochastic integrals. (In English). arXiv:1801. 07248 [math. PR]. 2018, 9 p. Available at: https://arxiv.org/abs/1801.07248
  172. Kuznetsov D. F. Theorems about integration order replacement in multiple Ito stochastic integrals. (In English). arXiv:1801. 04634 [math. PR]. 2018, 21 p. Available at: https://arxiv.org/abs/1801.04634
  173. Kuznetsov D. F. Expansion of multiple stochastic integrals according to martingale Poisson measures and according to martingales, based on generalized multiple Fourier series. (In English). arXiv:1801. 06501 [math. PR]. 2018, 20 p. Available at: https://arxiv.org/abs/1801.06501
  174. Kuznetsov D. F. Expansions of multiple Stratonovich stochastic integrals from the Taylor-Stratonovich expansion, based on multiple trigonometric Fourier series. Comparison with the Milstein Expansion. (In English). arXiv:1801. 08862 [math. PR]. 2018, 18 p. Available at: https://arxiv.org/abs/1801.08862
  175. Kuznetsov D. F. Expansion of multiple Stratonovich stochastic integrals of fifth multiplicity, based on generalized multiple Fourier series. (In English). arXiv:1802. 00643 [math. PR]. 2018, 21 p. Available at: https://arxiv.org/abs/1802.00643
  176. Kuznetsov D. F. To numerical modeling with strong orders 1. 5 and 2. 0 of convergence for multidimensional dynamical systems with random disturbances. (In Russ. ). arXiv:1802. 00888 [math. PR]. 2018, 15 p. Available at: https://arxiv.org/abs/1802.00888
  177. Kuznetsov D. F. Explicit one-step strong numerical methods of order 2. 5 for Ito stochastic differential equations, based on the unified Taylor-Ito and Taylor-Stratonovich expansions. (In English). arXiv:1802. 04844 [math. PR]. 2018, 23 p. Available at: https://arxiv.org/abs/1802.04844
  178. Kuznetsov D. F. Numerical simulation of 2. 5-set of multiple Ito stochastic integrals of multiplicities 1 to 5. (In English). arXiv:1805. 12527 [math. PR]. 2018, 16 p. Available at: https://arxiv.org/abs/1805.12527
  179. Kuznetsov D. F. Numerical simulation of 2. 5-set of multiple Stratonovich stochastic integrals of multiplicities 1 to 5. (In Russ). arXiv: 1806. 10705 [math. PR]. 2018, 16 p. Available at: https://arxiv.org/abs/1806.10705
  180. Kuznetsov D. F. New representation of Levy stochastic area, based on Legendre polynomials. (In English). arXiv: 1807. 00409 [math. PR]. 2018, 16 p. Available at: https://arxiv.org/abs/1807.00409
  181. Kuznetsov D. F. Strong numerical methods of order 3. 0 for Ito stochastic differential equations, based on the unified stochastic Taylor expansions and multiple Fourier-Legendre series (In English). arXiv:1807. 02190 [math. PR]. 2018, 30 p. Available at: https://arxiv.org/abs/1807.02190
  182. Kuznetsov D. F. Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. Comput. Math. Math. Phys. 58: 7 (2018), 1058-1070
  183. Lotka A. J. Undamped oscillations derived from the law of mass action. J. Amer. Chem. Soc. 42: 8 (1920), 1595-1599
  184. Maghsoodi Y., Harris C. J. In-probubility approximation and simulation of nonlinear jump-diffusion SDE. IMA J. Math. Control Inform. 4 (1987), 65-92
  185. Maghsoodi Y. Mean-square efficient numerical solution of jump-diffusion SDE. 1994. Preprint OR72. Univ. of Southampton. 26 p
  186. Maruyama G. Continuous Markov processes and stochastic equations. Rend. Circ. Math. Palermo. 4 (1955), 48-90
  187. McKenna J., Morrison J. A. Moments and correlation functions of a stochastic differential equation. J. Math. Phys. 11 (1970), 2348-2360
  188. McKenna J., Morrison J. A. Moments of solutions of a class of stochastic differential equations. J. Math. Phys. 12 (1971), 2126-2136
  189. Merton R. C. Option pricing when underlying stock returns and discontinuous. J. Financial Economics. 3 (1976), 125-144
  190. Merton R. C. Continuous-time finance. Oxford; N. Y., Blackwell Publ., 1990. 453 p
  191. Mikulevicius R. On some properties of solutions of stochastic differential equations. Lietuvos Mat. Rink. 4 (1983), 18-31
  192. Mikulevicius R., Platen E. Time discrete Taylor approximations for Ito processes with jump component. Math. Nachr. 138 (1988), 93-104
  193. Mikulevicius R., Platen E. Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151 (1991), 233-239
  194. Milstein G. N. The probability approach to numerical solution of nonlinear parabolic equations. 1997. Preprint No. 380, WIAS. 29 p
  195. Milstein G. N., Platen E., Schurz H. Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35: 3 (1998), 1010-1019
  196. Milstein G. N., Tretyakov M. V. Stochastic numerics for mathematical physics. Berlin, Springer-Verlag Publ., 2004. xx+596 p
  197. Milstein G. N., Tretyakov M. V. Numerical integration of stochastic differential equations with nonglobally lipschitz coefficients. SIAM J. Numer. Anal. 43: 3 (2005), 1139-1154
  198. Milstein G. N., Tretyakov M. V. Numerical algorithms for forward-backward stochastic differential equations. SIAM J. Sci. Comput. 28: 2 (2006), 561-582
  199. Milstein G. N., Tretyakov M. V. Practical variance reduction via regression for simulating diffusions. Reseach Reports in Mathematics. Report No. MA-06-019, University of Leicester, 2006. 24 p
  200. Milstein G. N., Tretyakov M. V. Solving linear parabolic stochastic partial differential equations via averaging over characteristics. Reseach Reports in Mathematics. Report No. MA-07-009, University of Leicester, 2007. 26 p
  201. Newton N. J. An asymptotically efficient difference formula for solving stochastic differential equations. Stochastics. 19 (1986), 175-206
  202. Newton N. J. Asymptotically optimal discrete approximations for stochastic differential equations. In theory and applications of nonlinear control systems. Ed. C. Byrnes, A. Lindquist. Amsterdam, 1986, p. 555-567
  203. Newton N. J. Asymptotically efficient Runge-Kutta methods for a class of Ito and Stratonovich equations. SIAM J. Appl. Math. 51 (1991), 542-567
  204. Nyquist H. Thermal agittation of electric charge in conductors. Phys. Rev. 32 (1928), 110-113
  205. Obuhov A. M. Description of turbulence in Lagrangian variables. Adv. Geophis. 3 (1959), 113-115
  206. Petrovski I. G. Uber das Irrfahrtproblem. Math. Ann. 109 (1934), 425-444
  207. Pettersson R. The Stratonovich-Taylor expansion and numerical methods. Stoch. Anal. Appl. 10: 5 (1992), 603-612
  208. Philips H. B., Wiener N. Nets and Dirichlet problem. J. Math. Phys. 2 (1923), 105-124
  209. Platen E. A Taylor-Ito formula for semimartingales solving a stochastic differential equation. Springer Lecture Notes in Control and Inform. Sci. 36 (1981), 157-164
  210. Platen E. A generalized Taylor formula for solutions of stochastic differential equations. Sankhya. 44A (1982), 163-172
  211. Platen E. An approximation method for a class of Ito processes with jump component. Lietuvos Mat. Rink. 22 (1982), 124-136
  212. Platen E., Wagner W. On a Taylor formula for a class of Ito processes. Probab. Math. Statist. 3 (1982), 37-51
  213. Platen E. Zur zeitdiskreten Approximation von Itoprozessen. Diss. B., IMath. Akad. der Wiss. der DDR, 1984. Berlin
  214. Platen E. Higher-order weak approximation of Ito diffusions by Markov chains. Probab. Eng. Inform. Sci. 6 (1992), 391-408
  215. Platen E. On weak implicit and predictor-corrector methods. Math. Comput. Simulation. 38 (1995), 69-76
  216. Platen E. An introduction to numerical methods for stochastic differential equations. Acta Numerica. 8 (1999), 197-246
  217. Platen E., Bruti-Liberati N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  218. Pontrjagin L. S., Andronov A. A., Witt A. A. Statistische auffassung dynamischer systeme. Phys. Zeit. 6: (1934), 1-24
  219. Reshniak V., Khaliq A. Q. M, Voss D. A., Zhang G. Split-step Milstein methods for multi-channel stiff stochastic differential systems. Applied Numerical Mathematics. 89 (2015), 1-23
  220. Richardson J. M. The application of truncated hierarchy techniques in the solution of a stochastic linear differential equation. In Stochastic Processes in Mathematical Phisics and Engineering. Proc. Symp. Appl. Math. Ed. R. Bellman. Amer. Math. Soc. Providence RI. 16 (1964), 290-302
  221. Rossler O. E. An equation for continuous chaos. Phys. Lett. 57A (1976), 397-398
  222. Rumelin W. Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19 (1982), 604-613
  223. Ryden T., Wiktorsson M. On the simulation of iterated Ito integrals. Stoch. Processes and their Appl. 91 (2001) 151-168
  224. Schurz H. Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise. Stoch. Anal. Appl. 14 (1996), 313-354
  225. Schurz, H. Numerical regularization for SDEs: construction of nonnegative solutions. Dynam. Systems Appl. 5: 3, (1996), 323-351
  226. Shkurko I. O. Numerical solution of linear systems of stochastic differential equations. Numer. Methods Statist. Modeling. Collected Scientific Works. Novosibirsk, 1987. p. 101-109
  227. Smoluhovski M. V. Drei Vortrage uber Diffusion Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Zeit. 17 (1916), 557-585
  228. Strook D. W., Varadhan S. R. S. Multidimensional diffusion processes. Berlin, Springer Publ., 1979. 338 p
  229. Talay D. Convergence pour chaque trajectoire d'un scheme d'approximation des EDS. Computes Rendus Acad. Sci. Paris. Ser. I. Math. 295 (1982), 249-252
  230. Talay D. Efficient numerical schemes for the approximation of expectations of functionals of the solution of an SDE and applications. Springer Lecture Notes in Control and Inform. Sci. 61 (1984), 294-313
  231. Talay D., Tubaro L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8: 4 (1990), 483-509
  232. Ueno T. The diffusion satisfying Wentzell's boundary conditions and the Markov processes on the boundary. Proc. Japan Akad. 36 (1960), 533-538
  233. Van der Ziel A. Fluctuation phenomena in semi-conductors. London, Butterworths Scientific Publ., 1959, 168 p
  234. Wagner W., Platen E. Approximation of Ito integral equations. Preprint ZIMM Akad. Wiss. DDR. Berlin. 1978. 27 p
  235. Wagner W. Unbiased Monte-Carlo evaluation of certain functional integrals. J. Comput. Phys. 71 (1987), 21-33
  236. Wagner W. Monte-Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stoch. Anal. Appl. 6 (1988), 447-468
  237. Wolf J. R. Neue Untersuchungen uber die Periode der Sonnenflecken und ihre Bedeutung. Mit. Naturforsch. Ges. Bern. 255, (1852), 249-270
  238. Wright D. J. The digital simulation of stochastic differential equations. IEEE Trans. Automat. Control. AC-19 (1974), 75-76
  239. Wright D. J. Digital simulation of Poisson stochastic differential equations. Internat. J. Systems Sci. 6 (1980), 781-785
  240. Zahri M. Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution. Journal of Taibah University for Science. 8: 2 (2014), 186-198

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