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Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane

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Abstract

In this paper we study stochastic Volterra equations in a plane. These equations contain integrals with respect to fields of locally bounded variation and square-integrable strong martingales. We prove the existence and the uniqueness of solutions of such equations with locally integrable (in some measure) trajectories, assuming that the coefficients of equations possess the Lipschitz property with respect to the functional argument. We prove that a solution of a stochastic Volterra integral equation in a plane is continuous with respect to parameter.

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References

  1. A. A. Gushchin, “On the General Theory of Random Fields in a Plane,” Usp. Mat. Nauk 37(6), 53–74 (1982).

    MathSciNet  Google Scholar 

  2. M. L. Kleptsyna and A. Yu. Veretennikov, “On Strong Solutions of Stochastic Ito-Volterra Equations,” Teor. Ver. i Ee Prim., No. 12, 32–40 (1984).

  3. E. Pardoux and P. Protter, “Stochastic Volterra Equations with Anticipating Coefficients,” Ann. of Probab. 18(4), 1635–1655 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  4. A. M. Kolodii, “On Conditions for Existence of Solutions of Integral Equations with Stochastic Line Integrals,” Probab. Theory and Math. Stat., 405–422 (1994).

  5. A. A. Gushchin and Yu. S. Mishura, “Davis Inequalities and Gandy Decomposition for Two-Parameter Strong Martingales. 1,” Teor. Ver. iMat. Stat., No. 42, 27–35 (1990).

  6. M. Dozzi, “Twoparameter Stochastic Processes,” Math. Research 61, 17–43 (1991).

    MathSciNet  Google Scholar 

  7. C. Stricker and M. Yor, “Calcul Stochastique D/’ependant d’un Paramètre,” Z. Wahrscheinlichkeitstheor. verw. Geb. 45, 109–133 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  8. A.V. Skorokhod, “On the Measurability of Random Processes,” Teor. Ver. i Ee Prim. 25(1), 140–142 (1980).

    MATH  MathSciNet  Google Scholar 

  9. N. A. Kolodii, “Some Properties of Random Fields Related to Stochastic Integrals with Respect to Strong Martingales,” J. of Math. Sci. 137(1), 4531–4540 (2006).

    Article  MathSciNet  Google Scholar 

  10. L. Horvath, “Gronwall-Bellman Type Integral Inequalities in Measure Spaces,” J. Math. Anal. and Appl. 202(1), 183–193 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  11. I. I. Gikhman and A. V. Skorokhod, Controlled Random Processes (Naukova Dumka, Kiev, 1977) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Volgograd State University, Universitetskii pr. 100, Volgograd, 400062, Russia

    N. A. Kolodii

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  1. N. A. Kolodii
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Correspondence to N. A. Kolodii.

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Original Russian Text © N.A. Kolodii, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 2, pp. 20–32.

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Kolodii, N.A. Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane. Russ Math. 54, 16–27 (2010). https://doi.org/10.3103/S1066369X10020039

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  • DOI: https://doi.org/10.3103/S1066369X10020039

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