A note on the LaSalle-type theorems for stochastic differential delay equations
The author presents an improvement of results obtained in an earlier paper [J. Math. Anal. Appl. 236, No. 2, 350-369 (1999; Zbl 0958.60057)]. In the article \(n\)-dimensional stochastic differential delay equations are considered, which are of the form \[ dx(t)=f(x(t),x(t-\tau),t) dt + g(x(t),x(t-\tau),t) dB(t),\tag{1} \] where \(B(t)\) denotes \(m\)-dimensional Brownian motion. The main theorem is a stochastic version of the LaSalle theorem, providing criteria for the determination of the almost sure asymptotic behaviour of the solution of (1). The improvement concerns the assumptions on the coefficient functions. The local Lipschitz and local linear growth conditions on \(f\) and \(g\) are relaxed to local boundedness in the first two arguments and uniform boundedness in the last argument of \(f\) and \(g\), in addition the existence and uniqueness of a solution of (1) is required. The results can thus be applied to a larger class of equations. The proof of the theorem, some corollaries and an extension to the multiple delay case are given. Several examples are presented, demonstrating the usefulness of the results.
scientific article; zbMATH DE number 1763448