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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.
##### MSC:
 60J60 Diffusion processes 60G46 Martingales and classical analysis 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)