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The generalized Itô–Venttsel’ formula in the case of a noncentered Poisson measure, a stochastic first integral, and a first integral

Abstract

We deduce an analog of the Itô–Venttsel’ formula for an Itô system of generalized stochastic differential equations (GSDE) with noncentered measure on the basis of a stochastic kernel of an integral invariant. We construct a system of GSDE whose solution is a kernel of an integral invariant connected with a solution to GSDE with noncentered measure. We introduce the notion of a stochastic first integral of a system of GSDE with noncentered measure and find conditions under which a random function is a first integral of a given system of GSDE.

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Correspondence to E. V. Karachanskaya.

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Original Russian Text © E.V. Karachanskaya, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 1, pp. 99–122.

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Karachanskaya, E.V. The generalized Itô–Venttsel’ formula in the case of a noncentered Poisson measure, a stochastic first integral, and a first integral. Sib. Adv. Math. 25, 191–205 (2015). https://doi.org/10.3103/S1055134415030049

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Keywords

  • Itô–Venttsel formula
  • generalized stochastic differential equation
  • noncentered Poisson measure
  • kernel of an integral invariant
  • stochastic first integral