Recalling that, strong comparison principle is able to identify a unique viscosity solution in . In this below, we have an example having its interior being discontinuous. Is there any theory to justify its uniqueness?
Its counter part of exit problem is the following. With underlying process
and exit time the value functions is defined as
For this simple exit problem, a straightforward computation leads to an explicit value : With a decomposition of by
the value function can be expressed explicitly as
Indeed, one can check that is a generalized viscosity solution of (1). However, it is discontinuous inside , in particular at any point on the set .