SOLUTION APPROXIMATIONS FOR LYAPUNOV TYPE EQUATIONS ASSOCIATED WITH LINEAR STOCHASTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS AND COUNTABLY INFINITE MARKOV JUMPS

Abstract

This paper discuses solution properties of the Lyapunov-type equations (LEs), associated with a class of linear stochastic equations with unbounded coefficients and countably infinite Markov jumps. It is proved that the mild solutions of these LEs can be obtained as weak limits of sequences of strong solutions of certain approximating systems of LEs. Unlike the case of linear stochastic equations without jumps [4], where the LEs acts on Banach spaces of linear and bounded operators and the mild solutions are strong limits of approximating strong solutions, in our case the LEs are defined on Banach spaces of infinite sequences of operators and the strong convergence is replaced by the weak convergence.

Keywords

• Lyapunov type equations
• mild solution
• strong solution
• stochastic differential equations.