We consider three optimzation problems. First one is optimal stopping problem, second is linear programming motivated from first problem, and the last is the dual problem of the second one. We expect the value functions of all three problems are equal to each other under certain assumptions. Given filtered probability space , let be an 1-D Markov diffusion process with a generator , defined by

In the above is the space of test functions. Let be a collection of countably additive measures on bounded by 1. Also, we denote

Define adjoint operator by

is well defined in the above way, since

- The linear functional defined on by
can be extended to larger space consistently by Han-Banach Theorem using sup norm. (We shall find a sepcific extension?)

- Riesz Representation Theorem on implies there exists unique measure , such that

Given a function and a constant , we consider optimial stopping (OS) problem:

Here, we use to denote the collection of all -stopping times satisfying almost surely, and means . One can use Ito’s formula to obtain, for all

where

- (A1) .

Under assumption (A1), is a martingale, and for all by optional sampling theorem. This yields

Then, taking and using dominated convergence theorem, one obtains

Next we introduce two measures by

Obviously, . One can rewrite (1) by

This motivates following linear programing problem:

One can write the above (LP) in a standard form. To see that, we first rewrite the constraint by

or shortly

Hence,

where , , and . One can expect

Duality formulation suggests

Weak duality suggests

How about the strong duality?