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We cannot now speak of maximizing the value of the objective function, since this function is now known only in a probabilistic sense.

We can, however, maximize its expected value.

For a single stage we may define Af where the maximization is by choice of Af.

We thus have an optimal policy which maximizes the expected value of the objective function for a given Af.

If we consider a process in which the outcome of one stage is known before passage to the next, then the principle of optimality shows that the policy in subsequent stages should be optimal with respect to the outcome of the first.

Then Af, the maximization being over all admissible Af and the integration over the whole of stage space.