Stochastic Programming - Index of

124 STOCHASTIC PROGRAMMING
Table 1 Distribution of the interest rate on account A. All outcomes have
probability 0.5, and the outcomes in period 2 are independent of the outcomes
in period 1.
Period Outcome 1 Outcome 2
1 8 12
2 5 9
A
8% or 12%
fee 20
A
5% or 9%
fee 20
A
S
B
fee 10 fee 10
0 3
B
7% 5%
Figure 6 Simple investment problem with uncertain interest rates.
B
S
2.4 Stochastic Decision Trees
We shall now see how decision trees can be used to solve certain classes
of stochastic problems. We shall initiate this with a look at our standard
investment problem in Example 2.2. In addition, let us now assume that the
interest on account B is unchanged, but that the interest rate on account A
is random, with the previously given rates as expected values. Charges on
account A are unchanged. The distribution for the interest rate is given in
Table 1. We assume that the interest rates in the two periods are described
by independent random variables.
Based on this information, we can give an update of Figure 4, where we
show the deterministic and stochastic parameters of the problem. The update
is shown in Figure 6.
Consider the decision tree in Figure 7. As in the deterministic case, square
nodes are decision nodes, from which we have to choose between account A
and B. Circular nodes are called chance nodes, and represent points at which
something happens, in this case that the interest rates become known.
Start at the top. In stage 0, we have to decide whether to put the money