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Stochastic differential equations

Published online by Cambridge University Press:  24 October 2008

D. A. Edwards  and
J. E. Moyal
D. A. Edwards
Affiliation:
Oriel CollegeOxford
J. E. Moyal
Affiliation:
Department of MathematicsUniversity Of Manchester
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Extract

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 4 , October 1955 , pp. 663 - 677
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Chandrasekhar, S.Stochastic problems in physics and astronomy. Rev. mod. Phys. 15 (1943), 1–89.CrossRefGoogle Scholar
(2)Doob, J. L.The Brownian movement and stochastic equations. Ann. Math., Princeton (2), 43 (1942), 351–69.CrossRefGoogle Scholar
(3)Doob, J. L.The elementary Gaussian process. Ann. math. Statist. 15 (1944), 229–82.CrossRefGoogle Scholar
(4)Dunford, N.Uniformity in linear spaces. Trans. Amer. math. Soc. 44 (1938), 305–56.Google Scholar
(5)Fréchet, M.Sur les fonctionnelles bilinéaires. Trans. Amer. math. Soc. 16 (1915), 215–34.Google Scholar
(6)Gelfand, I.Abstrakte Funktionen und lineare Operatoren. Rec. Math., Moscou (Mat. Shorn.) (N.S.), 4 (1938), 235–84.Google Scholar
(7)Graves, L. M.Riemann integration and Taylor's theorem in general analysis. Trans. Amer. math. Soc. 29 (1927), 163–77.Google Scholar
(8)Hille, E.Functional analysis and semigroups (New York, 1948).Google Scholar
(9)Ince, E. L.Ordinary differential equations (London, 1927).Google Scholar
(10)Kakutani, S.Concrete representation of abstract (L)-spaces and the mean ergodic theorem. Ann. Math., Princeton (2), 42 (1941), 523–37.Google Scholar
(11)Karhunen, K.Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. fenn. AI, 37 (1937), 1–79.Google Scholar
(12)Lévy, P.Processus stochastiques et mouvement brownien (Paris, 1948).Google Scholar
(13)Loève, M. Fonctions aléatoires du second ordre (Appendix to (12)).Google Scholar
(14)Moyal, J. E.Stochastic processes and statistical physics. J.R. Statist. Soc. B, 11 (1949), 150–210.Google Scholar
(15)Orlicz, W.Beiträge zur Theorie der Orthogonalentwicklung. II. Studia Math. 1 (1929), 241–55.CrossRefGoogle Scholar
(16)Pettis, B. J.On integration in vector spaces. Trans. Amer. math. Soc. 44 (1938), 277–304.CrossRefGoogle Scholar
(17)Pettis, B. J.A note on regular Banach spaces. Bull. Amer. math. Soc. (1938), 420–8.CrossRefGoogle Scholar
(18)Saks, S.Theory of the integral, 2nd ed. (Warsaw, 1937).Google Scholar