We consider a specific diffusion control problem. We focus on setting where there is one risky asset and one riskless asset, though we will see that much of the analysis passes over to multiple assets.
Def. [The Merton Problem – Plant Equation] In the Merton problem you wish to optimise your long run consumption. You may invest your wealth in a bank account receiving riskless interest , or in a risky asset with value obeying the following SDE
where each is an independent standard Brownian motion.
Wealth obeys the SDE
You can control your rate of consumption at time and the number of stocks the risky asset at time . Also, we define to be the wealth in the risky asset at time .
Def. [The Merton Problem – Objective] Given the above plant equation, , the objective is to maximize the long-term utility of consumption
Here is a positive constant and is a concave increasing utility function. The set is the set of policies given initial wealth . Further, let be the optimal objective with the integral starting for time with .
Let’s write the wealth respect to integrals in and :
Ex 1. Show that
Ex 2. Show that
Ans 2. Notice if we shift time by , a factor comes out,
Ex 3. Show that the HJB equation for the Merton Problem can be written as
or, alternatively, as
Ans 3. Recall that informally the HJB equation is
Notice that if we apply Ito’s formula to we get that
Applying this to the above term gives as required
Ex 4. [Continued] Optimizing the HJB over , show that
Ans 4. Differentiating the HJB equation in [[cDP:MertonHJB]] wrt gives
Now rearrange for .
Ex 5. [Continued] Given , show that the HJB equation becomes
Ans 5. Substituting gives
Merton for CRRA Utility
We focus on the case of CRRA utility, that is:
for (Recall the discussion on utility functions, Section [Util]). Thus we wish to solve for
Ex 6. Show that
Ans 6. Here we note that having a policy for initial wealth is the same as having a policy of wealth and then multiplying each amount invested by :
Ex 7. Show that
for some position constant .
Ans 7. In , let and .
Ex 8. Show that
Ans 8. Trival.
Ex 9. Show that if we define
then for a CARA utility
Ans 9. Differentiating gives now rearrange and substitute.
Ex 10. Show that, when optimizing over , the HJB equation is optimized by
which since , gives the required formed.
Ex 11. [Merton:CARA4] Show that
Ex 12. [Continued]Show that
Ans 12. By  and 
Ex 13. Given the optimal choices of and , Show that the HJB equation to be satisfied it must be that
To summarize: we notice we have show that the parameters given by
in exercises [4, 10, 13] give a solution to the HJB equation for the Merton problem. (Although we have not yet proven them to be optimal.)
We now give rigourous argument for the optimality of parameters , and for the Merton problem with CRRA utility. (This section can be skipped if preferred.)
Ex 14. Show that
(Hint: is concave.)
Ans 14. Since is concave we have that . Thus
Ex 15. Verify that
is a positive local martingale. [Hint: apply Ito’s formula to
Ex 16. [Continued] Show that
[Hint: Doob’s Martingale Convergence Theorem.]
Ans 16. Recall from stochastic integration theory that every positive local martingale is a supermartingale. Doob’s Martingale Convergence Theorem applied to  gives
Ex 17. [Continued] By direct calculation show that
[Hint: apply Fubini’s Theorem and note that
Ex 18. Now show that
Ans 18. Combining  and , we see that
Applying this to  we see that
as required. Thus is optimal.
The last exercise shows that the portfolio is optimal for the Merton problem with CRRA utility.