Stability of stochastic functional differential equations with respect to first approximation
Stochastic functional differential equations of the form (a) \( dx(t) = [(Vx)(t) + g(t)] dZ(t),\) where \(V\) is a certain type of linear operator, and (b) \( dx(t) = [(Nx)(t) + f(t)] dZ(t),\) where \(N\) is a nonlinear operator, are studied. \(Z\) is an \(m\)-dimensional semimartingale. Stability (with respect to the zero solution) of solutions of these equations is defined in terms of these solutions belonging (either locally or globally) to the following linear space (or certain subspaces of it): \[ {\mathcal M}_p^{\gamma} = \biggl\{ x : x \in D^n,\;\|x\|_{{\mathcal M}_p^{\gamma}} \overset{\text{def}}= \biggl( \sup_{t \geq 0} E|\gamma(t) x(t)|^p \biggr)^{1/p} < \infty\biggr\}, \] where \(D^n\) is a linear space of \(n\)-dimensional predictable stochastic processes. So essentially stability in the \(p\)th moment with decay rate \(\gamma(t)\) is considered, e.g. \(\liminf_{t \to \infty} \gamma(t) = \infty\) corresponds to asymptotic stability, and \(\gamma(t)= \exp(\beta t)\) to exponential stability. Assume that \(N\) can be expressed as \(Nx =Vx+Fx\), \(V\) linear and \(F\) nonlinear with \(\|Fx-Fy\|\leq \|x-y\|\) (in suitable norms), then theorems are presented concerning the local stability of solutions of equation (b) given the stability of solutions of (a). The theory is applied to an Itô equation with ``maximum''. Related results by J. Hale, R. Z. Khasminskij, V. Kolmanovskij, X. Mao, A. Myshkis, etc. are not mentioned.
scientific article; zbMATH DE number 1541542