- HJB equation for Merton Problem; CRRA utility solution; Proof of Optimality.
- Multiple Assets; Dual Value function Approach.
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 [cDP:MertonPlant]
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 .
Prop 1. The HJB equation for the Merton Problem can be written as Here the optimal is given by
Proof. First we note that we can rewrite the SDE for as follows:
Further note that if we shift the value function time by , a factor comes out,
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
Differentiating the HJB equation w.r.t. gives
Now rearrange for .
Merton for CRRA Utility
We focus on the case of CRRA utility, that is:
for . (See the discussion on utility functions, Section [Util].) Thus we wish to solve for
Prop 2. For a CRRA utility it holds that:
a) The Value function takes the form
for some position constant .
b) The HJB equation is optimized by
c) The HJB equation is satisfied by parameters
Proof. a) Note that having a policy for initial wealth is the same as having a policy of wealth and then multiplying each amount invested by :
Letting and gives the result.
b) By part a), and . So, by Prop [Mert:Prop],
which since , gives that . Further,
c) Applying a) and b) to the HJB equation in Prop 1 gives
Cancelling and rearranging gives the required for for .
To summarize: we notice we have shown that the parameters
give a solution to the HJB equation for the Merton problem. (Although we have not yet proven them to be optimal.)
We now give rigorous argument for the optimality of parameters , and for the Merton problem with CRRA utility. (This section can be skipped if preferred.)
Thrm 1. The parameters
are optimal for the Merton problem.
Proof. Since is concave, . Thus for we have that
Next we show that
is a positive local martingale. It is clear that the function is positive. Note that
and note that . Now lets apply Ito’s formula to . By Ito’s formula:
Now lets check terms.
Substituting these into Ito’s formula above gives,
Cancelling and rearrganging we get
is a local-Martingale. Recall from stochastic integration theory that every positive local martingale is a supermartingale.
Doob’s Martingale Convergence Theorem applied to gives
Since and by the definition of :
The last equality holds since and .
Combining the last equality and the inequality before that, we see that
Applying this to we see that
and, as required, is optimal.
Merton Portfolio Optimization with Multiple Assets
We now note how the above results extend to the case where there aren’t many assets. Now suppose that there are assets that can be in invested in. These obey the Stochastic Differential Equation
where is an independent Brownian motion for each .
Wealth now evolves according the SDE
where gives the amount of each asset held in the portfolio at time . Also we define as the wealth in each asset. As given in the definition of the Merton problem, our task is the maximize the objective
We now proceed through exercises that are very similar to the case with a single risky asset. We go through the proofs somewhat quickly.
Lemma 1. Show that the HJB equation for the Merton Problem can be written as
Proof. The proof follows more-or-less identically to Prop 1. Note that in this case Ito’s formula applied to gives
where Thus This is the drift term applied in the HJB equation. Thus recalling that
This gives the required HJB equation.
Lemma 2. Show the optimal asset portfolio in the HJB equation is given by
Proof. Considering Lemma 1 we have that
Solving for and substituting back into the maximization gives the answer.
Lemma 3. Show that for a CRRA utility the optimal solution to the HJB equation is given by
Proof. (In this proof when we refer to Prop 2 we mean that the argument which was applied in the single-asset setting is identical in the multiple asset setting.)
By Prop 2a)
for some constant . Differentiating twice gives
By Prop 2b), . Substituting these solutions into the HJB equation gives
Rearranging and solving for gives the required solution for .
Def [Merton Portfolio and Market Price Risk Vector] As given above,
is called the Merton Portfolio and
is called the Market Price Risk Vector.
Dual value function approach
We could solve the CRRA utility case because it had a special structure. We now give a method for solving for general utilities .
Here we assume that is continuous in and , concave in and satisfies
The HJB equation for the Merton problem is
We take the LF transform of ,
Further we define
where is such that .
The HJB equation can be written as
Moreover if we suppose that , for concave and increasing, the HJB equation becomes
Noticed in the first HJB equation above that we have got rid of the maximization and in the second we have a linear ODE, which can be solved using standard methods.
First we will show that
We can ignore the dependence of for the first two expressions i.e. take . Now , so
Now reintroducing dependence on ,
This gives the required derivatives in .
Substituting the expressions in , the HJB equation is
Now is we suppose that , for concave and increasing, then by the same argument as Prop 1a) we have that
Defining where is such that , the following are straightforward calculations:
where . Now substituting these terms into the HJB equation gives the result: