Geodesic
Two-Parameter Stochastic Volterra Equations
Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 525-537.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study two-parameter stochastic Volterra equations containing integrals over strong martingales and fields of bounded variation. For such equations, we prove the existence and uniqueness theorems for solutions with trajectories in the space of Borel functions which are locally square-integrable with respect to a locally finite measure. Under additional conditions on the coefficients of the equation, we prove the existence of a modification of the solution with trajectories continuous on the right and without discontinuities of the second kind.
Mots-clés : stochastic Volterra equation, martingale
Keywords: Borel function, metric separable space, stochastic integral, Hölder's inequality. 
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N. A. Kolodij. Two-Parameter Stochastic Volterra Equations. Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 525-537. http://geodesic.mathdoc.fr/item/MZM_2009_86_4_a4/

[1] A. A. Guschin, “K obschei teorii sluchainykh polei na ploskosti”, UMN, 37:6 (1982), 53–74 | MR | Zbl

[2] A. A. Guschin, Yu. S. Mishura, “Neravenstva Devisa i razlozhenie Gandi dlya dvuparametricheskikh silnykh martingalov. I”, Teoriya veroyatn. i matem. stat., 42 (1990), 27–35 | MR | Zbl

[3] M. Dozzi, “Two-parameter stochastic processes”, Stochastic Processes and Related Topics (Georgenthal, 1990), Math. Res., 61, Akademie-Verlag, Berlin, 1991, 17–43 | MR | Zbl

[4] M. L. Kleptsina, A. Yu. Veretennikov, “O silnykh resheniyakh stokhasticheskikh uravnenii Ito–Volterra”, TVP, 29:1 (1984), 154–158 | MR | Zbl

[5] É. Pardoux, P. Protter, “Stochastic Volterra equations with anticipating coefficients”, Ann. Probab., 18:4 (1990), 1635–1655 | DOI | MR | Zbl

[6] A. M. Kolodii, “On conditions for existence of strong and weak solution of stochastic Volterra equations”, Theory Stoch. Process., 2:3–4 (1996), 67–78

[7] C. Stricker, M. Yor, “Calcul stochastique dépendant d'un paramètre”, Z. Wahrsch. Verw. Gebiete, 45:2 (1978), 109–133 | DOI | MR | Zbl

[8] K. Kuratovskii, Topologiya, T. 1, Mir, M., 1966 | MR | Zbl

[9] I. I. Gikhman, A. V. Skorokhod, Upravlyaemye sluchainye protsessy, Naukova dumka, Kiev, 1977 | MR | Zbl

[10] K. Dellasheri, Emkosti i sluchainye protsessy, Mir, M., 1975 | MR | Zbl

[11] L. Horváth, “Gronwall–Bellman type integral inequalities in measure spaces”, J. Math. Anal. Appl., 202:1 (1996), 183–193 | DOI | MR | Zbl

[12] N. A. Kolodii, “Nekotorye svoistva sluchainykh polei svyazannykh so stokhasticheskimi integralami po silnym martingalam”, Veroyatnost i statistika. 8, Zap. nauchn. sem. POMI, 320, POMI, SPb., 2004, 80–96 | MR | Zbl