Most valuation problems in quantitative finance represent the desired value as the expected value of some random variable together with a stochastic model of market movements. Given the model, there are two ways to compute the expected value using evolution equations. One method starts with expected values known in the future and computes expected values at successively earlier times until the present expected values are found. The other method starts with given probabilities for current market conditions and works forwards in time to find probabilities for market conditions at a desired future time. These two evolution equations are similar but not identical. One of the differences is the natural direction of time change, backwards for expected values and forwards for probabilities.

This lecture discusses these evolution equations in the simple case of discrete time and discrete ``state space''. The main ideas are contained here. The extension to more complex situations, continuous time or state space, are mainly technical. The relationship between them is called ``duality''. It is an extension of the relationship between a matrix and its transpose.

Many financial instruments allow the holder to make decisions along the
way that effect the ultimate value of the instrument. American style
options, loans that be repaid early, and convertible bonds are examples.
To compute the value of such an instrument, we also seek the optimal
decision strategy. *Dynamic programming* is a computational method
that computes the value and decision strategy at the same time. It
reduces the difficult ``multiperiod decision problem'' to a sequence
of hopefully easier ``single period'' problems. It works backwards
in time much as the expectation method does. The tree method commonly
used to value American style stock options is an example of the
general dynamic programming method.

Tue Sep 15 17:12:32 EDT 1998