## Limit behavior of solutions of stochastic differential equations

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- by Avner Friedman PDF
- 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

## Additional Information

- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**170**(1972), 359-384 - MSC: Primary 60J60
- DOI: https://doi.org/10.1090/S0002-9947-1972-0378118-9
- MathSciNet review: 0378118