Dynamic restoration of the unknown function in the linear stochastic differential equation

TLDR: In this paper, a linear stochastic differential equation with diffusion is considered for portfolio selection in order to describe the time profile of the asset prices under risk investments, and an algorithm based on a combination of the methods of the theory of improperly posed problems and the theory with model is constructed in the class of finite-step algorithms counting on computer-aided realization.

Abstract: For the linear stochastic differential equation with diffusion, consideration was given to the problem of restoration of the unknown parameter characterizing the level of noise. Equations of this kind are used, in particular, in the problem of optimal portfolio selection in order to describe the time profile of the asset prices under risk investments. Restoration must be based on the measurements of the current phase state. The problem under study comes to the inverse problem of the ordinary matrix differential equation satisfied by the covariance matrix of the initial random process. The proposed algorithm which is based on a combination of the methods of the theory of improperly posed problems and the theory of positional control with model is constructed in the class of finite-step algorithms counting on computer-aided realization. The algorithm is stable to the errors in information and computations. The algorithm's precision was estimated in terms of the measurable realizations of the initial process.