The notion of certainty equivalence in stochastic dynammic programming
holds for linear quadratic utility functions. Does it also hold when
the cost function has linear constraints on the decision space?
For example, given the utility function
U(t) = (x(t)+y(t)) * e(t) - M*(y(t))^2
x(t+1) = x(t) + y(t)
U is the utility gained in period t, we choose y(t) to maxmize
expected cumulative utility over say T periods. And M is a constant.
e(t) is a random variable, for example it may be a mean reverting
process (e(t) = b * e(t-1) + i(t), where b is a constant between zero
and one, and i(t) is a random white noise term).
In the above problem, if e(t) is a random variable then since the
utility function is a quadratic, we can invoke certainty equivalence
and just deal with the mean of e(t). However, if there are linear
constraints on y(t), say y must be less than k, where k is some
constant, does certainty equivalence still hold? If it does, are there
any assumptions on the process generating e(t) that need to be made in
order for certainty equivalence to hold? |