<p>How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?</p>

<p>In Duffie's book "Dynamic Asset Pricing," he outlines the "Martingale method" of solving stochastic control problems. I won't reproduce the whole outline or notation here, but the the essentials are given on p.217 of his third edition book:</p>

<p><a href="https://i.sstatic.net/V1xO8.png" rel="noreferrer"><img src="https://i.sstatic.net/V1xO8.png" alt="enter image description here"></a>
<img src="https://i.sstatic.net/x67SK.png" alt="enter image description here">
<img src="https://i.sstatic.net/2PhTq.png" alt="enter image description here"></p>

<p>After some discussion of a generalization, he mentions the following (p.221):</p>

<blockquote>
  <p>Although this approach generates an explicit solution for the optimal-consumption
  policy up to an unknown scalar $\gamma$, it does not say much about the form of the optimal trading strategy, beyond its existence. The Notes cite sources in which an optimal strategy is represented in terms of the <em>Malliavin calculus</em>....</p>
</blockquote>

<p>I know how to solve for the optimal trading strategy using the Hamilton-Jacobi-Bellman approach, but I would like to learn how to do this using the Malliavin calculus and the Clark-Ocone theorem. Duffie's book does not give directions on how to do this. Does anyone know (or can reproduce here) the way in which we would derive the optimal trading strategy this way? (For an easy, clean demonstration, it would be nice to assume, say, 
$U(c) = E \int_0^\infty \frac{C^{1 - \gamma}}{1 - \gamma}$.)</p>

CC BY-SA 3.0
2015-12-07T08:57:58.383Z
I can't comment because I don't have enough reputation points, but have you found a solution to your problem? I can solve this problem using martingale methods when utility is CRRA. Is this what you are looking for? I don't know about solving the general problem using the Malliavan calculus.
pdevar
CC BY-SA 4.0
2022-10-13T07:21:24.503Z
Did you ever figure this out? Any references you could point people to?
TheIntern
CC BY-SA 4.0
2022-10-16T01:21:18.127Z
@TheIntern It's been a while, but I don't think I ever found anything. Sorry!
jmbejara
<finance><academic-graduate><portfolio-theory><stochastic-calculus><optimal-control>
How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?
jmbejara
# How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?

(Asked by: username_0 on 2015-02-06)

How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?

In Duffie's book "Dynamic Asset Pricing," he outlines the "Martingale method" of solving stochastic control problems. I won't reproduce the whole outline or notation here, but the the essentials are given on p.217 of his third edition book:

[![enter image description here](https://i.sstatic.net/V1xO8.png)](https://i.sstatic.net/V1xO8.png)
![enter image description here](https://i.sstatic.net/x67SK.png)
![enter image description here](https://i.sstatic.net/2PhTq.png)

After some discussion of a generalization, he mentions the following (p.221):

> Although this approach generates an explicit solution for the optimal-consumption
> policy up to an unknown scalar $\gamma$, it does not say much about the form of the optimal trading strategy, beyond its existence. The Notes cite sources in which an optimal strategy is represented in terms of the *Malliavin calculus*....

I know how to solve for the optimal trading strategy using the Hamilton-Jacobi-Bellman approach, but I would like to learn how to do this using the Malliavin calculus and the Clark-Ocone theorem. Duffie's book does not give directions on how to do this. Does anyone know (or can reproduce here) the way in which we would derive the optimal trading strategy this way? (For an easy, clean demonstration, it would be nice to assume, say,
$U(c) = E \int\_0^\infty \frac{C^{1 - \gamma}}{1 - \gamma}$.)

**Tags:** finance, academic-graduate, portfolio-theory, stochastic-calculus, optimal-control

> **Comments:**
> **username_1**: I can't comment because I don't have enough reputation points, but have you found a solution to your problem? I can solve this problem using martingale methods when utility is CRRA. Is this what you are looking for? I don't know about solving the general problem using the Malliavan calculus.  
> **username_2**: Did you ever figure this out? Any references you could point people to?  
> **username_0**: @username_2 It's been a while, but I don't think I ever found anything. Sorry!  

---

## 0 answers

