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The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four types of boundaries: exit, entrance, regular, and natural. For example, given a domain [l,u], where the diffusion begins at position x within [l,u], we can ask a plethora of questions regarding the boundary behavior at, say, l. Can a particle begun at x (for l < x < u) reach l in finite time? What kind of boundary is l (e.g. Is it a natural boundary?)? Etcetera. To answer these questions, one makes ample use of speed measures and scale functions, which are both necessarily time-independent. My question is given below.

Let l = l(t) and u = u(t) be time-dependent boundaries of a diffusion satisfying an Ito SDE with time-dependent drift and diffusion coefficients. The time dependence of the boundaries arises because the domain where the diffusion coefficient is positive changes with time. We assume that the diffusion coefficient is identically zero on the boundary itself. Is there a straightforward classification of boundary behavior for such a process utilizing, say, time-dependent scale functions (and potentially speed measures) or related ideas?

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  • $\begingroup$ Could you provide a bit more detail about the nature of the boundaries? Is their evolution deterministic or random but adapted to the filtration of the Ito process? Are the boundaries smooth functions of time? Are they well separated? $\endgroup$ Aug 21, 2016 at 15:44
  • $\begingroup$ Thanks for your question. The boundaries are deterministic functions of time. More specifically, they are the level sets for where the diffusion coefficient vanishes. We have the equation D(x,t) = 0, which is an implicit function for x = x(t). In the problems I have been investigating, there are two solutions, which I call l(t) and u(t), for lower and upper boundaries. The boundaries are smooth. For all times in the domain, u(t) > l(t), although in some instances, as time approaches T, l(t) approaches u(t). In that case, we have t is an element of [0,T). Thanks. $\endgroup$
    – Sam
    Aug 23, 2016 at 13:21

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