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On a probabilistic approach to the construction of the generalized diffusion processes. (English) Zbl 0988.60074
The author proposes a method for constructing a generalized diffusion process in the Euclidean space \(\mathbb R^d\) such that the drift vector is equal to \((q\nu+\alpha(x^S))\delta_S(x)\) and the diffusion matrix is equal to \(I+\beta\delta_S(x)\), where \(\nu\in\mathbb R^d\) is a given unit vector, \(S\) is a hyperplane in \(\mathbb R^d\) orthogonal to \(\nu\), \(x^S\) is the orthogonal projection of \(x\) on \(S\), \(q\in[-1,1]\) is a parameter, \(\alpha: S\to S\) is a measurable function, \(\beta\) is a nonnegative symmetric operator defined on \(S\), \(I\) is the identical operator on \(\mathbb R^d\) and \(\delta_S(x)\) is a generalized function. Processes of this type describe Wiener processes in \(\mathbb R^d\) with the Wentzell boundary conditions on the hyperplane \(S\) (partly reflection in the direction \(q\nu+\alpha(x^S)\) and the diffusion in \(S\) corresponding to the operator \(\beta\)). The problem of construction of such processes in a more general case using an analytic approach was considered by B. T. Kopytko and Z. J. Tsapovska [Theory Stoch. Process. 4(20), No. 1-2, 139-146 (1998; Zbl 0940.60079), Visn. L’viv. Univ., Ser. Mekh.-Mat. 56, 106-115 (2000; Zbl 0972.35157)], B. T. Kopytko [Theory Probab. Math. Stat. 59, 93-100 (1999); translation from Teor. Jmovirn. Mat. Stat. 59, 91-98 (1998; Zbl 0959.60069)] and by S. V. Anulova [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 25, 264-269 (1980; Zbl 0539.60077)] using a martingale formulation. In this paper the probabilistic approach to the problem is proposed and the process is constructed as a solution of a stochastic differential equation.
60J60 Diffusion processes
60G46 Martingales and classical analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)