Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local behaviour of solutions of stochastic integral equations

Author: William J. Anderson
Journal: Trans. Amer. Math. Soc. 164 (1972), 309-321
MSC: Primary 60H20
MathSciNet review: 0297031
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let X denote the solution process of the stochastic equation $ dX(t) = a(X(t))dt + \sigma (X(t))dW(t)$. In this paper, conditions on $ a( \cdot )$ and $ \sigma ( \cdot )$ are given under which the sample paths of X are differentiate at $ t = 0$ with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If $ \sigma ( \cdot )$ is Hölder continuous with exponent greater than $ \tfrac{1}{2}$ and $ a( \cdot )$ satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.

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

  • [1] D. A. Dawson, Equivalence of Markov processes, Trans. Amer. Math. Soc. 131 (1968), 1-31. MR 37 #5937. MR 0230375 (37:5937)
  • [2] I. V. Girsanov, On Ito's stochastic integral equation, Dokl. Akad. Nauk SSSR 138 (1961), 18-21 =Soviet Math. Dokl. 2 (1961), 506-509. MR 22 #10010. MR 0119244 (22:10010)
  • [3] K. Itô, Lectures on stochastic processes (notes by K. M. Rao), Tata Institute of Fundamental Research, Bombay, 1961. MR 759892 (86f:60049)
  • [4] H. P. McKean, Jr., Stochastic integrals, Probability and Math. Statist., no. 5, Academic Press, New York, 1969. MR 40 #947. MR 0247684 (40:947)
  • [5] A. V. Skorohod, Studies in the theory of random processes, Izdat. Kiev. Univ., Kiev, 1961; English transl., Addison-Wesley, Reading, Mass., 1965. MR 32 #3082a, b. MR 0185620 (32:3082b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60H20

Retrieve articles in all journals with MSC: 60H20

Additional Information

Keywords: Stochastic integral equations, sample path behaviour, differentiability of solution, uniqueness of solution
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society