Prof. Elena Karachanskaya
Far-Eastern State University, Russia
Title: Stochastic processes with invariants and their modeling
Abstract:
Models in the form of stochastic differential equations are being used to construct models of random processes more and more often. Such models arise in economics, financial mathematics, engineering, biology, etc. However, a more detailed consideration of the properties of a random process does not always allow constructing a model of the process under study.
An important condition for the existence of a dynamical system is the presence of invariants determining its most important properties in this system.
The paper will present a method for constructing models of a stochastic process for which a set of invariants is given. Models of stochastic processes are presented in the form of a system of Ito Stochastic differential equations with a Poisson measure. We will consider the case where a process can move from one subspace to another. A random process with switching at random moments of time will be presented. In all presented models, invariants will be preserved with probability 1.
Biography:
I'm a Doctor Sci. in Math. My research interests are
stochastic processes with invariants and their modeling with Ito's stochastic
differential equations (SDE).
My achievements:
·
multidimensional
Generalized Ito-Wentzel formula with Poisson processes – it’s construction,
several variants of conditions for the coefficients, and their proof,
·
introduction
of a Stochastic First Integral for systems of stochastic differential equations
with standard Poisson measure (GSDE),
·
method
of Construction of GSDE systems, on the basis of the set of invariants of this
system, or Stochastic First Integrals collection,
·
introduction
of the Programmed Control (PCP1), which allows the stochastic process described
in the form of GSDE, to keep the given invariant with probability 1,
·
I
am the author of the computer program for automatic construction of systems of
differential equations with given invariants.
My affiliation: Professor of the Department of
Information Systems and Technologies Far-Eastern State Transport Unversity,
Khabarovsk, Russia.
My education: Department of Mathematics at Tomsk State
University, Tomsk, Russia.
My most important publications:
1.
Chalykh E.: Constructing the set of program controls with probability 1 for one
class of stochastic
systems. Automation
and Remote Control.
- 2009. - Vol. 70. - No.8. - pp. 1364-1375.
2.
Karachanskaya E. V.: A Proof of
the Generalized Itˆo-Wentzell Formula via the Delta-Function and
the Density of
Normal Distribution. Yakutian
Mathematical Journal. – 2014. – Vol. 21. – No.3. – P.40–51.
3.
Karachanskaya E. V.: The generalized Itˆo-Venttsel’ formula in the case
of a non- centered Poisson measure, a stochastic first integral, and a first
integral.Sib. Adv. Math.. – 2015. – Vol.25.– No.191. – P. 191–205.
4.
Karachanskaya E. V.: A ”direct”
method to prove the generalized Itˆo-Venttsel’ formula for a generalized
stochastic differential equation. Sib. Adv. Math.. – 2016. - Vol.26. –
No.17. – P.17–29.
5.
Karachanskaya E.: Construction of programmed controls for a dynamic
system based on the set of its first integrals. J. Math. Sci.. – 2014. –
Vol.199. – No.5. – P.547–555.
6.
Averina T., Karachanskaya E., and Rybakov K.: Statistical analysis of
diffusion systems with invariants. Russ. J. Numer. Anal. Math. Model.
(RJMAMM). – 2018. - Vol.33. – is.1. – 13 p.
7.
Karachanskaya E. and Tagirova T.: Construction of stochastic transport
models with a constant function. IOP Conference Series Earth and
Environmental Science. - 2019. – Vol.403. – no. 012211. – P.13.
8.
Karachanskaya E. V.: Programmed control with probability 1 for
stochastic dy- namical systems. J. Math. Sci.. – 2020. – Vol.248. –
No.1. – P.67–79.
9.
Karachanskaya E. V.: Invariants for a dynamical system with strong
random per- turbations. Advances in Dynamical Systems Theory, Models,
Algorithms and Appli- cations. – 2021. – 21p.
10.
Doobko V., Karachanskaya E.: Application of an indicator random process
for modeling open stochastic systems. Mathematical
notes of NEFU. – 2024. – Vol.31. -No.2. – P.81–98.