Hi all,Here is the question: If an optimal stopping problem is formuated and solved as a free boundary problem, what are the conditions under which the free boundary problem has a unique solution? I am running into a situation where the optimal stopping problem can be solved using a pure probabilistic approach based on stochastic techniques, while when formulating the problem as a free boundary problem, the solution cannot be uniquely determined. I am wondering if there is some technical points one has to take into account when transforming an optimal stopping problem into a free boundary problem. Looking at the solution of the problem I have in my hand, the optimal stopping time obtained can be infinite with a probability 0<p<1. I do not know whether this is the cause for the non-uniqueness problem when solving the problem as a free boundary problem.best,Tw813

QuoteOriginally posted by: tw813Hi all,Here is the question: If an optimal stopping problem is formuated and solved as a free boundary problem, what are the conditions under which the free boundary problem has a unique solution? I am running into a situation where the optimal stopping problem can be solved using a pure probabilistic approach based on stochastic techniques, while when formulating the problem as a free boundary problem, the solution cannot be uniquely determined. I am wondering if there is some technical points one has to take into account when transforming an optimal stopping problem into a free boundary problem. Looking at the solution of the problem I have in my hand, the optimal stopping time obtained can be infinite with a probability 0<p<1. I do not know whether this is the cause for the non-uniqueness problem when solving the problem as a free boundary problem.best,Tw813What specific kind of free boundary problem are you trying to solve?

It is the problem of pricing an American call with negative interest rate rho which is negative enough so that rho+volatility^2/2 <0. If you try to solve this problem using the usual free boundary formulation, the unknowns cannot be solved uniquely. If you have a bit of time, please see the message I posted just before this message, titiled ''American call with negative interest rate...''.Best,TW813

Your assumption of negative interest rates violates first principals, so its no surprise you cant solve it using variational inequalities. Negative rates would mean negative stock price which would obviously never be in the exercise region.What is surprising is that you say it can be solved using a pure probabilitic approach. How have you done this?

Hi, thanks for your comments. I might have misunderstood your meaning, but a negative interest rate does not mean the stock price will be negative. Provided that the initial stock price is positive, a negative interest rate simply means the stock price will be drifted downward.