Overview
A discount factor is a number that is when multiplied by the vlaue of some payoff, returns the price of that payoff, that is, what a rational agent would pay in order to obtain the payoff in question. In the context of fixed income securities, the discount factor is used to value a security using prevent value analysis.
In the context of rational choice facing risk, a similar concept arises. Given a set of possible future states of the world, {% \omega_i %}, each with an associated payoff, the stochastic discount factor is a number (in this case, its actually a random variable) such that the following pricing equation holds.
{% p = \mathbb{E}[mx] %}
That is, the price is the
expectation
of the discount factor {% m %} times the payoff {% x %} in each possible state of the world.
Radon Nikodym Theorem
The stochastic discount factor is an instance of the Radon Nikodym Theorem
{% \nu(A) = \int_A f d \mu %}