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Published in:
The Merton Problem Read chapter Read first chapter

2013 | OriginalPaper | Chapter

3. Numerical Solution

Author : L. C. G. Rogers

Published in: Optimal Investment

Publisher: Springer Berlin Heidelberg

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Abstract

The third chapter of the book presents the main numerical methods that are useful in calculating solutions to the optimal control problems of earlier chapters when analytic methods fail. The main technique is policy improvement, but this requires an effective translation of an optimization problem for a controlled diffusion into an optimization problem for a controlled Markov chain, and various techniques for this are discussed.

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Footnotes
1
We explain the method assuming that the state variable is one-dimensional, but the methodology works also in higher dimensions.
 
2
For an infinite horizon problem, the value is not time dependent, so the time derivative \(\dot{V}\) of \(V\) does not appear.
 
3
The method works with the obvious changes also for discrete-time Markov chains.
 
4
In fact, exact maximization is not required in (3.5); finding some control which improves things will be enough provided we do not cycle round the algorithm not picking up enough gain. It is hard to state a sufficient set of conditions, but an inspection of the proof of Theorem 3.1 shows how the idea would work.
 
5
Of course, in two dimensions, we could apply the second smooth interpolation recipe to the situation considered first, just by turning the picture through \(45^\circ \), but this does not work in higher dimensions. The convex set we inscribed the circle into is the unit \(\ell ^1\)-ball in the first situation, with \(2d\) vertices in \(d\) dimensions, whereas in the second situation we are working with the unit \(\ell ^\infty \)-ball, with \(2^d\) vertices.
 
6
We could allow the coefficients of the SDE to depend on time, but for notational simplicity we eschew this apparent generality.
 
7
Quite possibly this condition is not needed; at the time of writing, this issue is not decided.
 
8
\(\ldots \) for simplicity assumed to exist and be unique \(\ldots \)
 
9
The non-CRRA utility example of Section 2.​8 is such an example.
 
10
In higher dimensions, the story is much more complicated.
 
11
Unique up to scalar multiples...
 
12
See Theorem V.50.7 in [34].
 
13
We explain the method assuming that the state variable is one-dimensional, but the methodology works also in higher dimensions.
 
14
For an infinite horizon problem, the value is not time dependent, so the time derivative \(\dot{V}\) of \(V\) does not appear.
 
Metadata
Title
Numerical Solution
Author
L. C. G. Rogers
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Optimal Investment

Print ISBN: 978-3-642-35201-0

Electronic ISBN: 978-3-642-35202-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-35202-7_3