Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

Abstract

We study optimal stopping problems for diffusion processes with discontinuous reward function. We give some results about the regularity of the value function and we show that, under suitable mild conditions on the underlying process, it has the same regularity of the reward function, namely, it is lower (respectively: upper) semicontinuous if the reward function is. The proofs for the two cases are quite different, and the upper semicontinuous case requires stronger conditions. Finally, we show that, in the case of lower semicontinuous reward, under suitable conditions the value function is a (discontinuous) viscosity solution of the associated variational inequalities.