Stability of solutions of stochastic differential equations with random delays
Stochastic functional differential equations of the form \( dx(t) = (\bar{V}x)(t)dZ(t),\) where \(\bar{V}\) is given by \((\bar{V}x)(t)=(\sum_{j=0}^{m_j}A_{1j}(t)x(h_{1j}(t)),\ldots, \sum_{j=0}^{m_m}A_{mj}(t)x(h_{mj}(t)))\), are studied. Here, \(A_{ij}\) are matrices, \(Z\) is an \(m\)-dimensional semimartingale and the memory-type arguments of \(x\) are predictable random processes. Stability (with respect to the zero solution) of solutions of these equations is defined in terms of these solutions belonging to the linear space \[ M_p^{\gamma} = \left\{ x : x \in D^n, \| x\| _{M_p^{\gamma}} := \left( \sup_{t \geq 0} E| \gamma(t) x(t)| ^p\right)^{1/p} < \infty\right\}, \] where \(D^n\) is a linear space of \(n\)-dimensional predictable stochastic processes. This notion of stability includes stability in the \(p\)th mean with decay rate \(\gamma(t)\), e.g. \(\liminf_{t \rightarrow \infty} \gamma(t) = \infty\) corresponds to asymptotic stability in the \(p\)th mean, and \(\gamma(t)= \exp(\beta t)\) to exponential stability in the \(p\)th mean. Two theorems are presented concerning \(M_{2p}^1\)- and \(M_{2p}^{1/\gamma}\)-stability of the zero solution of the above equation.
scientific article; zbMATH DE number 2176644