The following question remains unsolved, and post it here. Maybe it’s not that hard, maybe not well-posed.
Let be a complete probability space, be a standard Brownian Motion, and be the filtration generated by , augmented by the -null sets as usual. The financial market consists of two assets, a bank account with interest rate and a stock whose price follows the dynamics
Here the set of the admissible strategies is defined rigorously at below: Given , the set of admissible strategies, denoted by , is the space of -adapted processes over such that, for a.s. ,
- is RCLL and piecewise constant with finitely many jumps;
- , and .
Our main interest is the optimization under
Then, if is continuous, we have . But it’s not clear is open.
Let denote the space of functions satisfying:
- is uniformly Lipschitz continuous in and continuous in ;
- is continuous in for , and and exist. Moreover, if , then ; if , then .
- is a viscosity subsolution, if for any smooth function with at with , we have
- is a viscosity supersolution, if for any smooth with at , we have
- is a viscosity solution if it is both a viscosity supersolution and a viscosity subsolution.
We may want to characterize by unique viscosity solution with some other conditions.