Some New Aspect of Lyapunov Type Theorems for Stochastic Difference Equations With Continuous Time

Abstract

Some new Lyapunov type theorems for stochastic difference equations with continuous time are proven. It is shown that these theorems simplify an application of Lyapunov functionals construction method.KeyWords: Lyapunov type theorems, stochastic difference equations, stability, method of Lyapunov functionals construction.Stability investigation of hereditary systems [1-3] is connected often with construction of some appropriate Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and developed in [4][5][6][7][8][9][10][11] for both stochastic differential equations with aftereffect and stochastic difference equations with discrete time. After some modification of the basic Lyapunov-type stability theorem, this method was also used for stochastic difference equations with continuous time [12][13][14], which are popular enough in researches [15][16][17][18][19][20]. Here some new aspect of Lyapunov type theorems is shown, which allows to simplify an application of the general method of Lyapunov functionals construction for stochastic difference equations with continuous time. The theorems obtained here can similarly be applied for stochastic differential equations and stochastic difference equations with discrete time. I. DEFINITIONS AND BASIC LYAPUNOV TYPE THEOREMLet {Ω, F, P} be a probability space, {F t , t ≥ t 0 } be a nondecreasing family of sub-σ-algebras of F, i.e. 1 2 t t ⊂ F F for t 1 < t 2 , E be the expectation with respect to the measure P, E t = E(./F t ) be the conditional expectation with respect to σ-algebra F t .Consider the stochastic difference equation( ( ) ( ) ( ) ) ( ) x t h a t x t x t h x t h a t x t x t h x t h t h t t hwith the initial conditionHere x ∈ R n , h 0 , h 1 , … are positive constants, the functionals a 1 ∈ R n and a 2 ∈ R n×m satisfy the conditionA solution of problem (1), (2) is an F t -measurable process x(t) = x(t; t 0 , φ), which is equal to the initial function φ(t) from (2) for t ≤ t 0 and with probability 1 defined by (1) for t > t 0 .