In this talk, I will present a stochastic representation for an elliptic integro-differential type of free boundary problems that arises from solving perpetual optimal stopping for a relatively general payoff function of Levy processes.
The representation is unique and is expressed in terms of a Fourier integral operators generated by the running infimum or supremum of the Levy process killed at independent exponential random time. The boundary and the solution of the free boundary problems are obtained using the stochastic representation. In particular, the solution is shown to coincide with the optimal solution of the original optimal stopping problem and is found to be general enough to bring together the existing results already appeared in literature.
We show that the C1 smooth pasting condition holds if and only if the optimal boundary is regular for the interior points of the stopping regions in the sense to be made precise in the talk. In the case where the boundary is not optimal, we observe two interesting facts which will be illustrated by some numerical simulations.