Numerical Integration of Stochastic Differential EquationsSpringer Science & Business Media, 1994年11月30日 - 172 頁 This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations. Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with. This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory. |
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A-stable A.V. Skorokhod a(tx additive noises Cauchy problem coefficients computation consider convergence deterministic dw(s error Euler's method expansion EY(T function h₁ h²² implicit methods implies independent Isir Itô integrals Itô's formula Kalman-Bucy filter La(t Lemma linear Lipschitz condition mathematical expectation method of order modeling Monte-Carlo method numerical integration O(h¹ O(h² O(h³ obtain one-step approximation order of accuracy order of smallness p₁ p₂ partial derivatives proof prove random variables rectangle method relations respect result righthand side Runge-Kutta method Russian satisfies scalar solution step stochastic differential equations sufficiently Suppose system of stochastic systems with additive t+h t+h Theorem 1.1 third order ti+1 tk+1 trapezium method tx+1 vector weak approximation Wiener integrals Wiener processes X(tk x(to Xk+1 Xt,x Xx+1 მე მთ