Limit behavior of solutions of stochastic differential equations

Friedman, Avner

doi:10.1090/S0002-9947-1972-0378118-9

Limit behavior of solutions of stochastic differential equations
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Trans. Amer. Math. Soc. 170 (1972), 359-384 Request permission

Abstract:

Consider a system of $m$ stochastic differential equations $d\xi = a(t,\xi )dt + \sigma (t,\xi )dw$. The limit behavior of $\xi (t)$, as $t \to \infty$, is studied. Estimates of the form $E|\xi (t) - \bar \sigma w(t){|^2} = O({t^{1 - \delta }})$ are derived, and various applications are given.

References

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 353–367. MR 0047272
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
I. I. Gikhman and A. V. Skorokhod, Introduction to the theory of random processes, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by Scripta Technica, Inc. MR 0247660
I. I. Gihman and A. V. Skorohod, Stokhasticheskie differentsial′nye uravneniya, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR 0263172
G. L. Kulīnīč, The limit distribution behavior of the solution of a stochastic diffusion equation, Teor. Verojatnost. i Primenen 12 (1967), 548–551 (Russian, with English summary). MR 0215365
G. L. Kulinič, Limiting behavior of distributions of a solution of the stochastic diffusion equation, Ukrain. Mat. Ž. 19 (1967), no. 2, 119–125 (Russian). MR 0211467
G. L. Kulinič, Asymptotic normality of the distribution of the solution of a stochastic diffusion equation, Ukrain. Mat. Ž. 20 (1968), 396–400 (Russian). MR 0232441
G. L. Kulīnīč, Limit distributions of a solution of the stochastic diffusion equation. , Teor. Verojatnost. i Primenen 13 (1968), 502–506 (Russian, with English summary). MR 0239671
Paul Lévy, Le mouvement brownien, Mémor. Sci. Math., no. 126, Gauthier-Villars, Paris, 1954 (French). MR 0066588
Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. MR 253426, DOI 10.1002/cpa.3160220304