Optimal portfolio policy in a bond market

We have a small investor who wants to optimize his expected utility of terminal wealth over a finite time horizon. We allow him to invest in a money market account or a bond market which consists of a zero coupon bond existing from time zero to time of one half the investment time horizon, and another zero coupon bond existing from time of one half the investment time horizon to the end of the investment time horizon. To solve this problem, a two stage dynamic programming problem in continuous time is formulated and the corresponding Hamilton-Jacobi-Bellman equations of Dynamic Programming are solved. In particular, we determine the optimal value function, optimal portfolio process, and the corresponding wealth process. We state and prove a verification theorem which our candidate optimal value function and portfolio process are shown to satisfy. We analyze the behavior of our optimal portfolio process, optimal value function, and the associated wealth process. Finally, we compare our results where the bond market consists of one zero coupon bond. The resulting optimal value function and wealth process are the same as the two stage case, while the resulting optimal portfolio process agrees with the two stage case on the latter half of the investment time horizon.