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Invariance for stochastic equations with regular coefficients. (English) Zbl 0876.60034

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)

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
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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
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