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On the support of the solutions of stochastic differential equations. (English) Zbl 0837.60055

Theory Probab. Appl. 39, No. 3, 519-523 (1994) and Teor. Veroyatn. Primen. 39, No. 3, 649-653 (1994).
This paper proves a support theorem for the following class of stochastic differential equations \[ dx= b(t, x, m) dt+ \sum \sigma_i(t, x, m)\circ dm^i, \] where \(m\) is a continuous semimartingale. The coefficient function \(b\) is assumed to be locally bounded, continuous in \((x, m)\), and satisfies a monotonicity condition, \(\sigma\) is assumed to be \(C^1\), and satisfying a Lipschitz condition uniformly in bounded sets. The semimartingale \(m\) is absolutely continuous, satisfying some growth conditions together with its derivative. Under these conditions, the SDE admits a unique, strong solution. The support theorem is proved relative to the family of ODE’s, where \(m\) is replaced by a set of absolutely continuous functions that contains \(C^\infty\).
Reviewer: W.Kliemann (Ames)

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G30 Continuity and singularity of induced measures
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