Editing Stochastic programming (section)

==== Stagewise independent random process ====

For a general distribution of the process <math>\xi_t</math>, it may be hard to solve these dynamic programming equations. The situation simplifies dramatically if the process <math>\xi_t</math> is stagewise independent, i.e., <math>\xi_t</math> is (stochastically) independent of <math>\xi_1,\dots,\xi_{t-1}</math> for <math>t=2,\dots,T</math>. In this case, the corresponding conditional expectations become unconditional expectations, and the function <math>Q_t(W_t)</math>, <math>t=1,\dots,T-1</math> does not depend on <math>\xi_{[t]}</math>. That is, <math>Q_{T-1}(W_{T-1})</math> is the optimal value of the problem

:<math>
\begin{array}{lrclr}
\max\limits_{x_{T-1}}   & E[U(W_T)]    &   \\
\text{subject to} & W_T   &=&    \sum_{i=1}^{n}\xi_{iT}x_{i,T-1} \\
                    &\sum_{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\
		    & x_{T-1}     &\geq& 0
\end{array}
</math>

and <math>Q_t(W_t)</math> is the optimal value of

:<math>
\begin{array}{lrclr}
\max\limits_{x_{t}}   & E[Q_{t+1}(W_{t+1})]    &   \\
\text{subject to} & W_{t+1}   &=&    \sum_{i=1}^{n}\xi_{i,t+1}x_{i,t} \\
                    &\sum_{i=1}^{n}x_{i,t}&=&W_{t}\\
		    & x_{t}     &\geq& 0
\end{array}
</math>

for <math>t=T-2,\dots,1</math>.