Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The density function of the solution of a two-point boundary value problem containing small stochastic processes

Author: Ning Mao Xia
Journal: Quart. Appl. Math. 46 (1988), 29-47
MSC: Primary 34B15; Secondary 34E05, 34F05, 35R99, 60H10
DOI: https://doi.org/10.1090/qam/934679
MathSciNet review: 934679
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns a two-point boundary value problem for an $ m$thorder system of ordinary differential equations containing a vector stochastic process

$\displaystyle \xi \left( {t, \omega } \right) = {\xi _0}\left( t \right) + \var... ...ht) + {\varepsilon ^2}{\xi _2}\left( {t, \omega } \right) + \cdot \cdot \cdot .$

When $ \varepsilon $ is small, the existence and the asymptotic properties of the solution can be obtained by means of the shooting method, and its density function can be determined by solving a sequence of first-order deterministic partial differential equations.

References [Enhancements On Off] (What's this?)

  • [1] L. Arnold, Stochastic differential equations: Theory and applications, John Wiley & Sons, New York (1974) MR 0443083
  • [2] W. E. Boyce and Ning-Mao Xia, The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients, Quart. Appl. Math. 40, 419-445 (1983) MR 693876
  • [3] W. H. Fleming, Stochastic control for small noise intensities, SIAM J. Control 9, 473-517 (1971) MR 0304045
  • [4] I. I. Gihman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York (1972) MR 0346904
  • [5] R. Z. Khazminskii, On stochastic processes defined by differential equations with a small parameter, Theory Probab. Appl. 11, 211-228 (1966) MR 0203788
  • [6] M. D. Lax, The method of moments for linear random boundary value problems, SIAM J. Appl. Math. 31, 62-83 (1976) MR 0405790
  • [7] R. F. Pawula, Generalizations and extensions of the Fokker-Planck-Kolmogorov equations, IEEE Trans. Inform. Theory, IT-13, 33-41 (1967) MR 0216570
  • [8] T. T. Soong, Random differential equations in science and engineering, Academic Press, New York (1973) MR 0451405
  • [9] B. S. White and J. N. Franklin, A limit theorem for stochastic two-point boundary-value problems of ordinary differential equations, Comm. Pure Appl. Math. 32, 253-275 (1979) MR 512421
  • [10] Ning-Mao Xia, W. E. Boyce, and M. R. Barry, Two-point boundary value problems containing a finite number of random variables, Stochastic Anal. Appl. 1, 117-137 (1983) MR 700359
  • [11] Ning-Mao Xia and W. E. Boyce, The density function of the solution of a random initial value problem containing small stochastic processes, SIAM J. Appl. Math. 44, 1192-1209 (1984) MR 766197
  • [12] Ning-Mao Xia, The solutions for the two-point boundary value problems of stochastic differential equations containing small white noises, Acta Math. Appl. 8, 340-350 (1985) MR 843408

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Additional Information

DOI: https://doi.org/10.1090/qam/934679
Article copyright: © Copyright 1988 American Mathematical Society

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