Decomposition and stability of linear systems with multiplicative noise (Q1061087)

This paper deals with linear stochastic systems of the form \(dX=a(t)Axdt+\sum B_ ix\circ dW_ i\), where ''\(\circ ''\) denotes the symmetric stochastic integral. The explicit solvability of these equations depends on the structure of the Lie algebra generated by \(\{A,B_ 1,...,B_ m\}\), see the work of \textit{A. J. Krener} and \textit{C. Lobry} [Stochastics 4, 193-203 (1981; Zbl 0452.60069)] for a general treatment of these problems. Once an explicit solution is obtained, stability properties (such as pth-mean, pth-order stability) can be discussed. The stability criteria are applied to an abelian and a solvable example.