Numerical analysis of dynamics of oscillatory stochastic systems

Abstract

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( · ).

An introduction to numerical methods for stochastic differential equations

“…In a number of applications, especially those associated with multiplicative noise, the balanced implicit method showed much better stability behaviour than other methods: see, for instance, Schurz (1996a) and Fischer and Platen (1998). Implicit schemes or different concepts of numerical stability have been suggested and studied in a variety of papers, and we again mention a long list, including Talay (19826, 1984), Klauder and Petersen (1985), Pardoux and Talay (1985), Milstein (1988aMilstein ( , 1995a, Artemiev and Shkurko (1991), Drummond and Mortimer (1991), Kloeden and Platen (1992), Spigler (1992, 1993), Artemiev (1993aArtemiev ( , 1993&, 1994, Saito and Mitsui (19936), Hofmann and Platen (1994), Milstein and Platen (1994), Komori and Mitsui (1995), Hofmann and Platen (1996), Saito and Mitsui (1996), Schurz (1996a), Schurz (1996c), Ryashko and Schurz (1997), Burrage (1998), Higham (1998) and Petersen (1998). Despite all this work, stochastic numerical stability remains an open and challenging area of research.…”

Some issues in discrete approximate solution for stochastic differential equations

“…Applying above ways of resolution, we are studying a Rosenbrock-type numerical schemes ( [2]) of high weak order. Our result will be forthcoming.…”

“…Let $y(t_{n},\omega)$ and $y_{n}(\omega)$ be the exact and approximate solutions, respectively, for the event $\omega$ at time $t_{n}$ . When the equality $y(t_{n},\omega)=y_{n}(\omega)$ holds, the quantity $\delta_{n+1}(\omega)=y(t_{n+1},\omega)-y_{n+1}(\omega)$ (2) is called the local truncation error of the numerical scheme for $\omega$ . Furthermore, if the estimation in terms of conditional expectation…”