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On one-dimensional Markov SDEs

  • Part I: General Theory
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Stochastic Differential Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 43))

Abstract

In [1] the existence of strong solutions of one-dimensional SDEs with coefficients depending only on the present was established. The proof was based on the Ito change-of-variables formula applied to a specially chosen function. However that function was not sufficiently smooth to enable the application of the Ito formula immediately.

The aim of this paper is (1) to get Ito's formula for "bad" functions including the function of [1] (simolar results were obtained in [3] and also mentioned can also be obtained without the Ito formula from purely deterministic lemmas. In fact, it is seen from those lemmas that the specific properties of the Wiener process are more or less unessential in the proof.

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References

  1. R.J. Chitashvili; T.A. Toronjadze, On one-dimensional SDEs with unit diffusion coefficient. Structure of solutions, Stochastics, 4 (1981), 281–315.

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  2. M.P. Ershov, On non-anticipating solutions of stochastic equations, Proc. Third USSR-Japan Sympos on Probability Theory, August 1975, Lecture Notes in Math., Vol 550.

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  3. A.T. Wang, Generalized Ito's formula and additive functionals of Brownian motion, Z.Wahrscheinlichkeitstheorie verw.Gebiete, 41 (1977), 153–159.

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  4. T. Yamada; Sh. Watanabe, On the uniqueness of solutions of SDEs, J. Math. Kyoto Univ., 11 (1971), 155–167.

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Authors and Affiliations

  1. University of Essen-GHS, 4300, Essen 1, Germany-West

    M. P. Ershov & Kl. Gooßen

Authors
  1. M. P. Ershov
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  2. Kl. Gooßen
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M. Kohlmann N. Christopeit

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© 1982 Springer-Verlag

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Ershov, M.P., Gooßen, K. (1982). On one-dimensional Markov SDEs. In: Kohlmann, M., Christopeit, N. (eds) Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol 43. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044287

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  • DOI: https://doi.org/10.1007/BFb0044287

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12061-2

  • Online ISBN: 978-3-540-39518-8

  • eBook Packages: Springer Book Archive

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