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: This paper concerns a two-point boundary value problem for an $m$thorder system of ordinary differential equations containing a vector stochastic process \[ \xi \left ( {t, \omega } \right ) = {\xi _0}\left ( t \right ) + \varepsilon {\xi _1}\left ( {t, \omega } \right ) + {\varepsilon ^2}{\xi _2}\left ( {t, \omega } \right ) + \cdot \cdot \cdot .\] 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.
- Ludwig Arnold, Stochastic differential equations: theory and applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR 0443083
- William 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 (1982/83), no. 4, 419–445. MR 693876, DOI https://doi.org/10.1090/S0033-569X-1983-0693876-9
- Wendell H. Fleming, Stochastic control for small noise intensities, SIAM J. Control 9 (1971), 473–517. MR 0304045
- Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR 0346904
- R. Z. Has′minskiĭ, Stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen 11 (1966), 240–259 (Russian, with English summary). MR 0203788
- Melvin D. Lax, The method of moments for linear random boundary value problems, SIAM J. Appl. Math. 31 (1976), no. 1, 62–83. MR 405790, DOI https://doi.org/10.1137/0131007
- R. F. Pawula, Generalizations and extensions of the Fokker-Planck-Kolmogorov equations, IEEE TRans. Information Theory IT-13 (1967), 33–41. MR 0216570, DOI https://doi.org/10.1109/tit.1967.1053955
- T. T. Soong, Random differential equations in science and engineering, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Mathematics in Science and Engineering, Vol. 103. MR 0451405
- 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 (1979), no. 2, 253–275. MR 512421, DOI https://doi.org/10.1002/cpa.3160320203
- Ning Mao Xia and William E. Boyce, Two-point boundary value problems containing a finite number of random variables, Stochastic Anal. Appl. 1 (1983), no. 1, 117–137. MR 700359, DOI https://doi.org/10.1080/07362998308809006
- Ning Mao Xia and William E. Boyce, The density function of the solution of a random initial value problem containing small stochastic processes, SIAM J. Appl. Math. 44 (1984), no. 6, 1192–1209. MR 766197, DOI https://doi.org/10.1137/0144085
- Ning Mao Xia, Solutions for two-point boundary value problems of stochastic differential equations containing a small white noise, Acta Math. Appl. Sinica 8 (1985), no. 3, 340–350 (Chinese, with English summary). MR 843408
L. Arnold, Stochastic differential equations: Theory and applications, John Wiley & Sons, New York (1974)
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)
W. H. Fleming, Stochastic control for small noise intensities, SIAM J. Control 9, 473–517 (1971)
I. I. Gihman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York (1972)
R. Z. Khazminskii, On stochastic processes defined by differential equations with a small parameter, Theory Probab. Appl. 11, 211–228 (1966)
M. D. Lax, The method of moments for linear random boundary value problems, SIAM J. Appl. Math. 31, 62–83 (1976)
R. F. Pawula, Generalizations and extensions of the Fokker-Planck-Kolmogorov equations, IEEE Trans. Inform. Theory, IT-13, 33–41 (1967)
T. T. Soong, Random differential equations in science and engineering, Academic Press, New York (1973)
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)
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)
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)
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)
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Article copyright:
© Copyright 1988
American Mathematical Society