Milian, Anna Invariance for stochastic equations with regular coefficients. (English) Zbl 0876.60034 Stochastic Anal. Appl. 15, No. 1, 91-101 (1997). A random process \(X(t)\), where \(t\in J\) and \(J\) is an interval in \(\mathbb{R}\), is said to be viable in \(K\), a subset of \(\mathbb{R}^m\), if \(P[X(t)\in K\), \(t\in J]=1\). A set \(K\subset \mathbb{R}^m\) is invariant by the Itô equation \[ X(t)=x+\int^t_0 f(X(s))ds+\int^t_0 g(X(s)) dW(s)\tag{*} \] if every solution \(X(\cdot)\) to (*), starting at any \(x\in K\), is viable in \(K\) on \([0,\infty)\). Using the idea of \(r\)-contingent cones, the author provides a criterion for the invariance of each closed subset of \(\mathbb{R}^m\). A similar criterion is also provided using Stratonovich rather than Itô equations. Reviewer: A.Dale (Durban) Cited in 6 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations Keywords:Ito equations; Stratonovich equations; contingent cones; invariance PDFBibTeX XMLCite \textit{A. Milian}, Stochastic Anal. Appl. 15, No. 1, 91--101 (1997; Zbl 0876.60034) Full Text: DOI References: [1] Aubin J.P., Differential Inclusions (1984) · Zbl 0538.34007 [2] Aubin J.P., Annali Scuola Normale di Pisa 27 pp 595– (1990) [3] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1981) · Zbl 0495.60005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.