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Stochastic differential inclusions. (English) Zbl 0911.93049

A stochastic differential inclusion is formulated in terms of stochastic differentials of continuous semimartingales. In particular, concepts of strong and weak solutions of the inclusion \[ dx_t\in F(t,x_t)dt+G(t,x_t)dw_t \] are introduced. Here \(F,G:[0,1]\times R^n\to \text{Comp} (R^n)\) are Borel measurable set-valued mappings. It is assumed that both \(F\) and \(G\) are bounded in Hausdorff metric by a scalar function from \(L^2([0,1])\) almost everywhere. The term \(w_t\) on the right-hand side of the differential inclusion is a one-dimensional Wiener process starting at zero.
Several results concerning the existence of solutions to such differential inclusions are formulated. For example, it is proved that a weak solution exists if both \(F\) and \(G\) are convex-valued, lower semi-continuous and grow linearly bounded in \(x\). For the existence of a strong solution, the requirement of a linear growth estimate has to be replaced by a condition on Lipschitz continuity.
Reviewer: D.Silin (Berkeley)

MSC:

93E03 Stochastic systems in control theory (general)
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
34A60 Ordinary differential inclusions
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