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Stochastic viability and a comparison theorem. (English) Zbl 0820.60041

The author considers the stochastic differential Itô equation in \(\mathbb{R}^ m\), \[ X(t) = x^ 0 + \int^ t_{t_ 0} f(s,X(s)) ds + \int^ t_{t_ 0} g(s, X(s)) dW(s), \quad t \geq t_ 0, \] where \(x^ 0 \in \mathbb{R}^ m\), \(f : [0, + \infty) \times \mathbb{R}^ m \to \mathbb{R}^ m\), \(g : [0,+\infty) \times \mathbb{R}^ m \to \mathbb{R}^{mr}\) and \(W\) is the \(r\)- dimensional Wiener process. The set \(K\) has the stochastic viability property with respect to the pair \((f,g)\) if for any \(x^ 0 \in K\) and any \(t_ 0 \geq 0\) every solution \(X\) satisfies \[ P\{X(t) \in K,\;t \geq t_ 0\} = 1. \] The main result of the paper is a criterion of stochastic viability for a convex polyhedron \(K\) and some comparison theorems.

MSC:

60H20 Stochastic integral equations
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