Stability of strong solutions of stochastic differential equations. (English) Zbl 0673.60065

Stochastic integral equations with respect to semimartingales are considered. Sufficient conditions under which a sequence of driving processes converges in law (or in probability) is given. This stability theorem is proved under assumptions both for the driving processes and the semimartingales. Some important applications are presented. Here the conditions are less strict than those in the existing literature.
Reviewer: Sv.Gaidov


60H20 Stochastic integral equations
60F17 Functional limit theorems; invariance principles
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[1] Aldous, D., A concept of weak convergence for stochastic processes viewed in the Strasbourg manner, (1979), Statistical Laboratory University of Cambridge, Preprint
[2] Billingsley, P., Convergence of probability measures, (1968), John Wiley and Sons New York · Zbl 0172.21201
[3] Dellacherie, C.; Meyer, P.A., Probabilités et potentiel, (1980), Hermann Paris
[4] Doleans-Dade, C., On the existence and unicity of solutions of stochastic differential equations, Z. warsch. verw. geb., 36, 93-101, (1976) · Zbl 0343.60038
[5] Emery, M., Stabilite des solutions des equations differentielles stochastiques: application aux integrales multiplicatives stochastiques, Z. warsch. verw. geb., 41, 241-262, (1978) · Zbl 0351.60054
[6] Emery, M., Equations differentielles lipschitziennes: etude de la stabilite, () · Zbl 0408.60064
[7] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam · Zbl 0495.60005
[8] Jacod, J., Calcul stochastique et problemes de martingales, () · Zbl 0414.60053
[9] Jacod, J., Convergence en loi de semimartingales et variation quadratique, () · Zbl 0458.60037
[10] Jacod, J.; Shiryayev, A.N., Limit theorems for stochastic processes, (1987), Springer-Verlag Berlin
[11] Jakubowski, A., On the Skorokhod topology, Ann. inst. Henri poincare B, 22, 263-285, (1986) · Zbl 0609.60005
[12] A. Jakubowski, J. Memin and G. Pages, Convergence en loi des suites d’integrales stochastiques sur l’espace D^{1} de Skorokhod, to appear in Probab. Th. Rel. Fields. · Zbl 0638.60049
[13] Kushner, H., Introduction to stochastic control, (1971), Holt, Rinehart and Winston, Inc New York · Zbl 0293.93018
[14] Liptser, R.; Shiryayev, A.N., Theory of matingales, (1986), Nauka Moscow, (in Russian)
[15] Mackevicius, V., S^{p} stabilite des solutions d’equations differentielles stochastiques avec semimartingales directrices discontinues, C.R. acad. sc. Paris 302, 19, 689-692, (1986), Serie I · Zbl 0587.60045
[16] Mackevicius, V., Sp stability of symmetric stochastic differential equations with discontinuous driving semimartingales, Ann. inst. Henri Poincaré B, 23, 575-592, (1987) · Zbl 0636.60057
[17] Metivier, M.; Pellaumail, J., On a stopped Doob’s inequality and general stoch. equations, Ann. probab., 8, 96-114, (1980) · Zbl 0426.60059
[18] Meyer, P.A.; Zheng, W.A., Tightness criteria for laws of semimartingales, Ann. inst. Henri Poincaré B, 20, 353-372, (1984) · Zbl 0551.60046
[19] Protter, P., On the existence, uniqueness, convergence and explosions of solutions of s.d.e., Ann. probab., 5, 243-261, (1977) · Zbl 0363.60044
[20] Protter, P., Hp stability of solutions of stochastic differential equations, Z. warsch. verw. geb., 44, 337-372, (1978)
[21] Protter, P., Approximations of solutions of s.d.e. driven by semimartingales, Ann. probab., 13, 716-743, (1985) · Zbl 0578.60055
[22] Rebolledo, R., Central limit theorems for local martingales, Z. warsch. verw. geb., 51, 269-286, (1980) · Zbl 0432.60027
[23] Słomiński, L., Necessary and sufficient conditions for extended convergence of semimartingales, Probability and math. statistics, 7, 77-93, (1986) · Zbl 0622.60038
[24] Słomiński, L., Approximation of predictable characteristics of processes with filtrations, () · Zbl 0694.60035
[25] Stricker, C., Lois de semimartingales et criteres de compacite, () · Zbl 0558.60005
[26] Stroock, D.W.; Varadhan, S.R.S., Multidimensional diffusion processes, (1979), Springer-Verlag Berlin
[27] Wong, E.; Zakai, M., On the convergence of ordinary integrals to stochastic integrals, Ann. math. statist., 36, 1560-1564, (1965) · Zbl 0138.11201
[28] Yamada, K., A stability theorem for stochastic differential equations and application to stochastic control problems, Stochastics, 13, 257-279, (1984) · Zbl 0553.60055
[29] Yamada, K., A stability theorem for stochastic differential equations with application to storage processes, random walks and optimal stochastic control problems, Stoch. processes appl., 23, 199-220, (1986) · Zbl 0609.60068
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