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

Kurenok, V. P.; Lepeyev, A. N.

Rocky Mt. J. Math. 38, No. 1, 139-174 (2008).

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.

MSC:

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

Keywords:

multi-dimensional stochastic differential equations; locally integrable coefficients; Krylov’s estimates; Wiener process; weak convergence

Citations:

Zbl 0974.60062; Zbl 0583.60052