The Cauchy problem for systems of stochastic differential equations is being solved. Numerical methods have been suggested for integration of the systems of the general form which can be considered as a generalization of the Rosenbrock and similar techniques for solving systems of ordinary differential equations. A convergence theorem has been proved. Integration schemes have been designed for linear systems of stochastic differential equations which possess an arbitrary order of compatibility both for the additive and the multiplicative noises. Numerical results are given.The statistic simulation of trajectories is at present the only technique of analysis of the dynamics of oscillatory stochastic systems practically in all real situations. It concerns nonlinear models that arise in analysing oscillations of structures, evaluating the reliability of parts and devices in random vibrations, estimating the stability of vehicle movement and its control systems and in analysing self-sustained oscillating regimes in radio engineering. In view of such particular state of numerical methods of solution of stochastic differential equations with oscillating solutions, greater requirements are imposed on them. And first of all the user of the methods must be sure that such probabilistic characteristics as an average amplitude and a medium frequency of oscillations of the solution to stochastic differential equations are estimated to a required accuracy. The paper is devoted to construction of special families of numerical methods of solution of stochastic differential equations, which are the most convenient for analysing stochastic oscillatory systems. CAUCHY PROBLEM FOR SYSTEM OF STOCHASTIC DIFFERENTIALEQUATIONS. BASIC DEFINITIONS Let {ß,^P} be a complete probabilistic space. For each t > 0 by <^c & denote a sub-a-algebra of events observed up to and including the time moment t.Let us give some definitions [5].(1) If ^ c ^ for ^ < t 2 and ^= ^+ 0 , where ^+ 0 = n /i>0^+ / z > then the totality of sub-a-algebras {^} is called a flow.(2) The random process y( · ) is consistent with the flow {^} if y(t) is â -measurable random vector for each t > 0.(3) The homogeneous Gaussian process with independent increments such that = 0 < W(t)W T (s) > = / · min(i,j) for t > 0, s > is called an Af-dimensional standard Wiener process W( · ).
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