×
Tudor, Constantin

On Volterra equations driven by semimartingales. (English) Zbl 0649.60072

J. Differ. Equations 74, No. 2, 200-217 (1988).
Stochastic equations of the form \[ X_ t=H_ t+\int^{t}_{0}F(t,s,X)dZ_ s \] are considered where Z is a finite- dimensional semimartingale, H is a cadlag process and F is some functional. The reasoning is based on P. Protter’s transformation rule [Ann. Probab. 13, 519-530 (1985; Zbl 0567.60065)], on a Gronwall type inequality and a domination property for semimartingales proved in the paper.
Existence and pathwise uniqueness of strong solutions of Volterra equations driven by semimartingales are established. The present results mark a certain progress in the existing literature. Finally convergence of some approximating processes to the solution is considered.
Reviewer: Sv.Gaidov

Cited in 2 Documents

MSC:

60H20 Stochastic integral equations
45D05 Volterra integral equations

Keywords:

cadlag process; Existence and pathwise uniqueness; strong solutions of Volterra equations driven by semimartingales; convergence of some approximating processes

Citations:

Zbl 0567.60065
PDFBibTeX XMLCite
\textit{C. Tudor}, J. Differ. Equations 74, No. 2, 200--217 (1988; Zbl 0649.60072)
Full Text: DOI

References:

[1] Berger, M.; Mizel, V., Volterra equations with Ito integrals, J. Integral Equations, 2, 187-245 (1980) · Zbl 0442.60064
[2] Berger, M.; Mizel, L. V., Volterra equations with Ito integrals, II, J. Integral Equations, 2, 319-337 (1980) · Zbl 0452.60073
[3] Bichteler, K., Stochastic integration and \(L^p\)-theory of semimartingales, Ann. Probab., 9, 49-89 (1981) · Zbl 0458.60057
[4] Da Prato, G.; Iannelli, M.; Tubaro, L., Dissipative functions and finite dimensional stochastic differential equations, J. Math. Pures Appl., 57, 173-180 (1978) · Zbl 0394.60066
[5] Dellacherie, C.; Meyer, P. A., Probabilities and Potential, B: Theory of Martingales (1980), North-Holland: North-Holland Amsterdam
[6] Doleans-Dade, C.; Meyer, P. A., Equations différentielles stochastiques, (Sém. Probab. XI, Lecture Notes in Math., 581 (1977)) · Zbl 0374.60080
[7] Emery, M., Stabilité des solutions des équations différentielles stochastiques: Application aux intégrales multiplicatives stochastiques, Z. Wahrsch. Verw. Gebiete, 41, 241-262 (1978) · Zbl 0351.60054
[8] Emery, M., Equations différentielles stochastiques lipschitziennes; Étude de la stabilité, (Sém. Probab. XIII, Lecture Notes in Math., 721 (1979)) · Zbl 0408.60064
[9] Gihman, I. V.; Skorohod, A. V., Stochastic Differential Equations and Applications (1982), Naukova Dumka: Naukova Dumka Kiev, [in Russian] · Zbl 0242.60003
[10] Gyongy, I.; Krylov, N. V., On stochastic equations with respect to semimartingales I, Stochastics, 4, 1-21 (1980) · Zbl 0439.60061
[11] Ito, I., On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type, Kodai Math. J., 2, 158-170 (1979) · Zbl 0421.60056
[12] Jacod, J.; Memin, J., Weak and strong solutions of stochastic differntial equations: Existence and stability, Lecture Notes in Math., 851 (1981)
[13] Kleptsyna, M. L.; Veretennikov, A. Ju, On the strong solutions of Ito-Volterra stochastic equations, Theory Probab. Appl., 29, 154-158 (1984), [in Russian] · Zbl 0542.60059
[14] Kravets, T. N., On the existence and uniqueness of solutions of stochastic integral equations of Volterra type, Theory of Random Processes, 8 (1980), [in Russian] · Zbl 0461.60080
[15] Kravets, T. N., Weak solutions of stochatic integral equations of Volterra type, Theory of Random Processes, 9, 55-60 (1981), [in Russian] · Zbl 0478.60072
[16] Kolodii, A. M., On the existence of solutions of stochastic Volterra integral equations, Theory of Random Processes, 11, 51-57 (1983), [in Russian] · Zbl 0542.60060
[17] McShane, E. J., Stochastic differential equations, J. Multivariate Anal., 5, 121-177 (1975) · Zbl 0323.60059
[18] Mel’nikov, A. V., Stochastic equations and Krylov’s estimates for semimartingales, Stochastics, 10, 81-102 (1983) · Zbl 0539.60059
[19] Metivier, M.; Pellaumail, J., Stochastic Integration (1980), Academic Press: Academic Press New York · Zbl 0426.60059
[20] Pellaumail, J., Solutions faibles et semimartingales, (Sém. Probab. XV, Lecture Notes in Math., 850 (1981)) · Zbl 0468.60057
[21] Pratelli, M., Majoration dans \(L^p\) du type Metivier-Pellaumail pour les semimartingales, (Sém. Probab. XVII, Lecture Notes in Math., 986 (1983)) · Zbl 0508.60047
[22] Protter, P., On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations, Ann. Probab., 5, 243-261 (1977) · Zbl 0363.60044
[23] Protter, P., Right-continuous solutions of systems of stochastic integral equations, J. Multivariate Anal., 7, 204-214 (1977) · Zbl 0363.60045
[24] Protter, P., \(H^p\)-stability of solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete, 44, 337-352 (1978) · Zbl 0369.60072
[25] Protter, P., Volterra equations driven by semimartingales, Ann. Probab., 13, 519-530 (1985) · Zbl 0567.60065
[26] Protter, P., A comparison of stochastic integrals, Ann. Probab., 7, 276-289 (1975) · Zbl 0404.60062
[27] Tudor, C., Successive approximations for solutions of stochastic integral equations of Volterra type, J. Math. Anal. Appl., 103, 27-37 (1984) · Zbl 0598.60064
[28] Tudor, C., Sur les solutions fortes des équations différentielles stochastiques, C. R. Acad. Sci. Paris, 299, no. 4, 117-120 (1984) · Zbl 0583.60058
[29] Wy, R. Q.; Mao, X., Existence and uniqueness of the solutions of stochastic differential equations, Stochastics, 11, 19-32 (1983) · Zbl 0535.60054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.