Large-scale stochastic singularly perturbed systems (Q1122224)

A multiparameter - multi time scale singularly perturbed linear stochastic system is considered. It can be written in the form \[ (1)\quad \epsilon_ idx_ i=\sum^{n}_{j=1}D_ iA_{ij}(t,\epsilon)x_ jdt+\sqrt{\sigma_ i}D_ iF_ i(t,\omega)dW_ i, \] where \(A_{ij}(t,\epsilon)\), \(F_ i(t,\omega)\) are block matrices and \(D_ i\) are diagonal matrices. The objective of the paper is to use the solution process of a completely decoupled auxiliary singularly perturbed system in order to approximate the solution of (1) in the mean squared sense. The auxiliary system is derived using a successive decoupling scheme which consists of a slow and a fast mode decomposition and which brings the deterministic parts of (1) into diagonal form. An incorporation of the solution of this system together with a nonsingular linear transformation leads to a closed form expression for the first-order approximation of (1). It is also pointed out how the results in this paper are related to other approaches in the literature.