*Original article (link) posted: 22/09/2005*

**Topics in the class**

1) Conditional Probability and Feller Property

(a) The definitions of a

*transition function: q*and the

*conditional expectation operator: T*.

(b) The properties of

*Tg*(

*g*is a measurable function);

*measurable*,

*non-negative*(if

*g*non-negative), and

*bounded*(if

*g*bounded).

(c)

*Feller Property*.

2) Dynamic Programming Set Up

(d) The def. of a reward function and a feasibility correspondence.

(e) Example; Neo-classical growth model, Portfolio choice, Capital accumulation, Search, and, Price with inertia (menu cost).

(f) The def. of

*history*and

*policy*(action).

(g) Setting up optimization problem and value function.

3) Bellman (Optimality) Equation

(h) Necessity: If the value function

*V*is measurable, then

*TV=V*.

(i) Sufficiency: If the bounded and measurable function

*U*solves

*U=TU*, then

*U*is larger than or equal to

*V*. Additionally if there is a selection from the optimality equation, then

*U=V*.

*Note)*a

*selection*from

*TU=U*is a

*stationary Markovian policy*which solves

*TU=U*.

**Comments**

Basic concepts in Measure theory such as sigma-algebra and measurability are postulated. I should check what Feller Property exactly means. (I'm not sure if it's just a definition or with necessary and sufficient conditions.)

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