The value function of optimal stopping problem is related to Bellman equation of the type

where the operator is defined by

Suppose comparison principle [CP] holds for the viscosity solution, then the unique solution can be characterized by the pointwise infimum of the piecewise smooth continuous viscosity supersolutions.

To proceed, we first define the space of supersolution, for any open set

We extend a function of into by . If there is no confuse, we shall use in replace of . Also define the space of continuous piecsewise smooth supersolution by

Consider

**Proposition 1** * Suppose [CP] holds for (1) and is continuous, then is the unique viscosity solution of (1). *

*Proof:* First, one can show an arbitrary is a supersoluiton by the similar to the proof of Proposition 2.8 of [CC95]. Since is continuous, Lemma 4.5 of [CIL92] implies is again a supersolution.

Suppose is not the subsolution. Then, there exists and a parabola

that touches at from the above, satisfying

By continuity of , there exists such that

Consider

where

Then, we conclude

- on , since when . In particular, we have
- since

By strict inequality (3) and the definition (2) of , there exist two functions such that

Note that,

satisfies

which leads to a contradiction to the definition (2) of .

### Like this:

Like Loading...

So if the CP holds, the value of the dual is the unique viscosity solution to the equation $F[u](x)=0$?

I guess you are right.