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The behavior of solutions of stochastic differential inequalities
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  • Published: December 1995

The behavior of solutions of stochastic differential inequalities

  • S. Assing1 &
  • R. Manthey1 

Probability Theory and Related Fields volume 103, pages 493–514 (1995)Cite this article

  • 238 Accesses

  • 15 Citations

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Summary

LetX andZ be ℝd -valued solutions of the stochastic differential inequalities dX t ≤a(t,X t )dt+σ(t,X t )dW t andb(t, Z t )dt+σ(t, Z t )dW t ≤dZ t , respectively, with a fixed ℝm-valued Wiener processW. In this paper we give conditions ona, b and σ under which the relationX 0≤Z 0 of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like ‘maximal/minimal solution’ of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such ‘solutions’ as well as some Gronwall-type estimates.

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Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, D-07740, Jena, Germany

    S. Assing & R. Manthey

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  1. S. Assing
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  2. R. Manthey
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Assing, S., Manthey, R. The behavior of solutions of stochastic differential inequalities. Probab. Th. Rel. Fields 103, 493–514 (1995). https://doi.org/10.1007/BF01246336

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  • Received: 15 August 1994

  • Revised: 20 March 1995

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01246336

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Mathematics Subject Classification (1991)

  • 60H10
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