Fiabilitate si Durabilitate - Fiability & Durability Supplement no 1/ 2012 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 423 A NEW REPRESENTATION RESULT FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE MARKOV JUMPS AND MULTIPLICATIVE NOISE V.M. UNGUREANU, Constantin Brâncuşi University, Tg-Jiu, ROMANIA Abstract. In this paper we give a new representation of the conditional mean square of the solutions for a class of stochastic differential linear equations with infinite Markov jumps (SDELMs) and multiplicative noise. The obtained result is related to the solutions of two Lyapunov type differential equations defined on ordered Banach spaces of sequences of bounded operators. Keywords: seqences, matrix, subspace; 1. INTRODUCTION In the last decades, the SDELMs with and without multiplicative noise have attracted the interest of the researchers [5], [6] and led to new applications in modern queuing network theory [4] or in the study of safety-critical and high integrity systems (see [1] and the references therein.) As in the discrete time-case (see for e.g [9], [8]), the representation of the conditional mean square of the solutions for SDELMs play an important role in studying different stability and optimal control problems ([8], [5], [6], [1]). So, in this paper we establish a new representation result based on the solution properties of some Lyapunov type equations associated with the discussed SDELMs. 2. NOTATIONS Let Z be an interval of integers, which may be finite or infinite. Let n R be the n - dimensional Euclidian space of real numbers and let   R m n M  be the real normed linear space of all m n  matrices with real entries; if n m  we will write   R n M instead of   R n n M  . Let   Z R m n M l  be the space of all Z -sequences    Z     i m n i M g g } { R with the property that      i i g g Z Z sup : . It can be shown by using a standard procedure that   Z R m n M l  is a real Banach space when endowed with the usual term-wise addition, the real scalar multiplication and the norm Z . . The Banach subspace of   Z R n M l formed by all sequences  Z   i i g g } { of symmetric matrices  Z  i g i , will be denoted by   Z R n S l . An element   Z R n M l g  is said to be positive, and we write 0  g , iff  i g is a nonnegative matrix (  0  i g ) for all Z  i . If n I is the identity matrix from   R n M , then   ,... , , ... n n n I I I   is an element of   Z R n M l .