Stability Properties of Constrained Jump-Diffusion Processes

Details Stability Properties of Constrained Jump-Diffusion Processes Stability Properties of Constrained Jump-Diffusion Processes

2005-01-02

arxiv.org/abs/math/0501014v1

Electronic Journal of Probability, Vol. 7, paper 22, p. 1-31 (2002)

Mathematics

Probability

Scientific paper

We consider a class of jump-diffusion processes, constrained to a polyhedral cone $G\subset\R^n$, where the constraint vector field is constant on each face of the boundary. The constraining mechanism corrects for ``attempts'' of the process to jump outside the domain. Under Lipschitz continuity of the Skorohod map \Gamma, it is known that there is a cone \mathcalC such that the image \Gamma\phi of a deterministic linear trajectory \phi remains bounded if and only if \dot\phi\in\mathcalC. Denoting the generator of a corresponding unconstrained jump-diffusion by \cll, we show that a key condition for the process to admit an invariant probability measure is that for x\in G, \cll \id(x) belongs to a compact subset of \mathcalC^o.

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