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A comparison theorem for stability of linear stochastic implicit difference equations of index 1

In this paper we study linear stochastic implicit difference equations (LSIDEs for short) of index-1. We give a definition of solution and introduce an index-1 concept for these equations. The mean square stability of LSIDEs is studied by using the method of solution evaluation. An example is given to illustrate the obtained results. | VNU Journal of Science Mathematics Physics Vol. 36 No. 3 2020 24-31 Original Article A Comparison Theorem for Stability of Linear Stochastic Implicit Difference Equations of Index-1 Nguyen Hong Son1 2 Ninh Thi Thu1 1 Faculty of Mathematics Mechanics and Informatics VNU University of Science 334 Nguyen Trai Thanh Xuan Hanoi Vietnam 2 Faculty of Natural Science Tran Quoc Tuan University Son Tay Hanoi Vietnam Received 26 June 2020 Accepted 11 July 2020 Abstract In this paper we study linear stochastic implicit difference equations LSIDEs for short of index-1. We give a definition of solution and introduce an index-1 concept for these equations. The mean square stability of LSIDEs is studied by using the method of solution evaluation. An example is given to illustrate the obtained results. Keywords LSIDEs index solution mean square stability. 1. Introduction In this paper we consider the linear time-varying stochastic implicit difference equation of the form An X n 1 Bn X n Cn X n n 1 n 1.1 where An Bn Cn d d the leading coefficient An may be singular and n is a standard one- dimensional scalar random process. LSIDEs is generalization of linear stochastic difference equations which have been well investigated in the literature see 1-4 . They arise as mathematical models in various fields such as population dynamics economics electronic circuit systems or multibody mechanism systems with random noise see e.g. 5-8 . They can also be obtained from stochastic differential algebraic equations SDAEs by some discretization methods see 9-12 . In comparison with linear stochastic ________ Corresponding author. Email address nghson80@gmail.com https doi.org 10.25073 2588-1124 vnumap.4570 24 N.H. Son N.T. Thu VNU Journal of Science Mathematics Physics Vol. 36 No. 3 2020 24-31 25 difference equations LSIDEs present at least two major difficulties the first lies in the fact that it is not possible to establish general existence and uniqueness results due to their more complicate .