On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients. (English) Zbl 0728.60066

In the present paper the authors have studied the stochastic equations of the type \[ (*)\quad X_ t=x+\int^{t}_{0}B(X_ s)dM_ s+\int^{t}_{0}A(X_ s)d<M>_ s,\quad t\in R^+, \] where \(M=(M^ 1,...,M^ d)\) is a d-dimensional continuous local martingale and the coefficients A, B are noncontinuous. The results on the existence and weak convergence of the solutions of (*) are given under some suitable conditions on the functions involved in (*).


60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
60J60 Diffusion processes
Full Text: DOI


[1] Aldous, D. J., Stopping time and tightness, Ann. Probab., 6, 335-340 (1978) · Zbl 0391.60007
[2] Anulova, S.; Pragarauskas, H., On strong Markov weak solutions of stochastic equations, Liet. Mat. Rinkinys, XVII, 2, 5-26 (1977) · Zbl 0381.60053
[3] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[4] Engelbert, H. J.; Schmidt, W., On one-dimensional stochastic differential equations with generalized drift, (Proc. IFIP Working Conference. Proc. IFIP Working Conference, Marseille-Luminy 1984. Proc. IFIP Working Conference. Proc. IFIP Working Conference, Marseille-Luminy 1984, Lecture Notes in Control and Inform. Sci. No. 69 (1985), Springer: Springer Berlin-New York), 143-155, Stochastic differential systems · Zbl 0545.60060
[5] Jakubowski, A.; Mémin, J.; Pages, G., Convergence en los des suites d’intégrales stochastiques sur l’espace \(D}^1\) de Skorokhod, Probab. Theory Rel. Fields, 81, 111-137 (1989) · Zbl 0638.60049
[6] Krylov, N. V., The selection of a Markov process from a Markov system of processes, Izw. Akad. Nauk SSSR, 37, 691-708 (1973), [In Russian.]
[7] Krylov, N. V., Controlled Diffusion Processes (1982), Springer: Springer New York · Zbl 0514.93070
[8] Mel’nikov, A. V., Stochastic equations and Krylov’s estimates for semimartingales, Stochastics, 10, 81-102 (1983) · Zbl 0539.60059
[9] Métivier, M., Semimartingales (1982), de Gruyter: de Gruyter Berlin-New York
[10] Rebolledo, R., Central limit theorems for local martingales, Z. Wahrsch. Verw. Gebiete, 51, 269-286 (1980) · Zbl 0432.60027
[11] Rozkosz, A.; Słomiński, L., On weak convergence of solutions of one dimensional stochastic differential equations, Stochastics and Stochastic Reports, 31, 27-54 (1990) · Zbl 0706.60059
[12] Stroock, D. V.; Varadhan, S. R.S., Multidimensional Diffusion Processes (1979), Springer: Springer New York · Zbl 0426.60069
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.