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

We now introduce our optimization problem on a subinterval . Given and an admissible trading strategy , we shall set and . Then the wealth process satisfies

Then, our optimization problem is:

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

- .

Define

and

Then, if is continuous, we have . But it’s not clear is open.

For a smooth function , define For a smooth function , denote

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 .

Let .

- 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.

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