After turning the random duration to finite during, I have discovered some interesting facts:
1. With the discountinous boundary conditions, the value function V(t,x) is not C^{1,2}. c.f. Pham's book (example on singular control)
2. Need to study viscosity solution (totally forgotten subject)
3. Now that it's a finite boundary problem, it becomes how to replicate/superreplicate the digital option with short constraint on portfolios
4. c.f. Cvitanic or Uwe Wystup's dissertation from CMU.
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