Local behaviour of solutions of stochastic integral equations
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- by William J. Anderson PDF
- Trans. Amer. Math. Soc. 164 (1972), 309-321 Request permission
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
- Donald A. Dawson, Equivalence of Markov processes, Trans. Amer. Math. Soc. 131 (1968), 1–31. MR 230375, DOI 10.1090/S0002-9947-1968-0230375-4
- I. V. Girsanov, On Ito’s stochastic integral equation, Soviet Math. Dokl. 2 (1961), 506–509. MR 0119244
- K. Ito, Lectures on stochastic processes, 2nd ed., Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 24, Distributed for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by K. Muralidhara Rao. MR 759892
- H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
- A. V. Skorohod, Issledovaniya po teorii sluchaĭ nykh protsessov (Stokhasticheskie differentsial′nye uravneniya i predel′nye teoremy dlya protsessov Markova), Izdat. Kiev. Univ., Kiev, 1961 (Russian). MR 0185619
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 164 (1972), 309-321
- MSC: Primary 60H20
- DOI: https://doi.org/10.1090/S0002-9947-1972-0297031-9
- MathSciNet review: 0297031