On multi-dimensional SDEs with locally integrable coefficients. (English) Zbl 1161.60021

Summary: We consider the multi-dimensional stochastic equation \[ X_t=x_0+\int^t_0B(s,X_s)dW_s+\int^t_0A(s,X_s) ds \] where \(x_0\) is an arbitrary initial value, \(W\) is a \(d\)-dimensional Wiener process and \(B:[0,+\infty)\times \mathbb{R}^d\to\mathbb{R}^{d^2}\), \(A:[0,+\infty)\times\mathbb{R}^d\to\mathbb{R}^d\) are measurable diffusion and drift coefficients, respectively. Our main result states sufficient conditions for the existence of (possibly, exploding) weak solutions. These conditions are some local integrability conditions of coefficients \(B\) and \(A\). From one side, they extend the conditions from [H. J. Engelbert, V. P. Kurenok, Georgian Math. J. 7, No. 4, 643–664 (2000; Zbl 0974.60062)] where the corresponding SDEs without drift were considered. On the other hand, our results generalize the existence theorems for one-dimensional SDEs with drift studied by H. J. Engelbert and W. Schmidt [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 69, 143–155 (1985; Zbl 0583.60052)]. We also discuss the time-independent case.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J65 Brownian motion
60G44 Martingales with continuous parameter
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