Open AccessMULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS
Author(s) -
Z. I. Nagolkina,
Yuri Filonov
Publication year - 2021
Publication title -
prikladnaâ geometriâ i inženernaâ grafika
Language(s) - English
Resource type - Journals
ISSN - 0131-579X
DOI - 10.32347/0131-579x.2021.100.205-214
Subject(s) - mathematics , multiplicative function , stochastic differential equation , equivalence (formal languages) , expression (computer science) , operator (biology) , representation (politics) , stochastic process , mathematical analysis , pure mathematics , computer science , biochemistry , chemistry , statistics , repressor , politics , transcription factor , political science , law , gene , programming language
In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed.
Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole.
It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.