One-dimensional Wiener process with the properties of partial reflection and delay

In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than $1/2$.

ON OPTIMAL CONTROL OF A STOCHASTIC EQUATION WITH A FRACTIONAL WIENER PROCESS

In recent years, a new direction of research has emerged in the theory of stochastic differential equations, namely, stochastic differential equations with a fractional Wiener process. This class of processes makes it possible to describe adequately many real phenomena of a stochastic nature in financial mathematics, hydrology, biology, and many other areas. These phenomena are not always described by stochastic systems satisfying the conditions of strong mixing, or weak dependence, but are described by systems with a strong dependence, and this strong dependence is regulated by the so-called Hurst parameter, which is a characteristic of this dependence. In this article, we consider the problem of the existence of an optimal control for a stochastic differential equation with a fractional Wiener process, in which the diffusion coefficient is present, which gives more accurate simulation results. An existence theorem is proved for an optimal control of a process that satisfies the corresponding stochastic differential equation. The main result was obtained using the Girsanov theorem for such processes and the existence theorem for a weak solution for stochastic equations with a fractional Wiener process.