I am trying to solve a multi-period free boundary problem, of an Ornstein–Uhlenbeck process, where each stopping decision at each period adds a different constant (penalty or bonus).
Solving with a value function framework, which its change in time expectation equals zero, where the change in the value function over small time interval, $dt$, is approximated by a Taylor expansion (until order 2).
My problem is that in the multi-period framework adding the constant bonus (or penalty) may break the value function differentiable property and the value function may have a non differentiable point (the value function is still continuous). What are the consequences of using the Taylor (Ito) framewoork in context of the value function when the value function has a non differentiable point? Is there a well known measure or approximation to this error?
Thanks