The main reference paper is El Karoui's paper: "Optimal Investment Decisions When Time Horizon is Uncertain".
Now the value function is associated with a terminal (utility) function g(u)=Id_{u=D}.
I don't know whether the HJB equation has a C^{1,2} solution.
Very difficult to find the close form solution of PDE.
I'm also finding that the recent paper http://www.maths.univ-evry.fr/prepubli/308.pdf could be a useful tool to link the problem to BSDE or FBSDE. Maybe I shall call my advisor?
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