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Published in:
The Merton Problem Read chapter Read first chapter

2013 | OriginalPaper | Chapter

2. Variations

Author : L. C. G. Rogers

Published in: Optimal Investment

Publisher: Springer Berlin Heidelberg

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Abstract

The second chapter of the book studies a wide range of different examples which are all in some sense variations on the basic Merton examples of Chapter 1. We study what happens when preferences change; or asset dynamics are changed; or objectives are changed.

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previous chapter The Merton Problem
next chapter Numerical Solution
Footnotes
1
As a check of the effect of the assumed boundary conditions, I calculated the efficiency at \(r=0\), which came out to 0.6972201 using a 7-standard deviation grid, and a 9-standard deviation grid, and a 5-standard deviation grid.
 
2
Compared with the Merton problem where we assume that \(r = \bar{r}\).
 
3
... at least in the case \(R>1\) which we deal with here.
 
4
Recall (2.3) that we are using \(R=2\).
 
5
The reason we do not allow \(\alpha =1\) is that the dependence on \(q\) is linear, and the problem degenerates; in effect, in this situation it is always possible to transfer an amount of wealth directly into \(\xi \) by a delta-function transfer, so the problem is degenerate.
 
6
We slightly abuse notation here; \(r_\rho (x,v)\) is a function of the difference \(v-x\) only, so we write \(r_\rho (z)\) for \(r_\rho (0,z)\).
 
7
Of course, we have \(V(w) = -K\) for all \(w <0\).
 
8
Combinations of the two cases could be considered, where the function \(\sigma \) takes more than 1 value, but fewer than \(N\). This could be handled by similar techniques, but we omit discussion as it is not particularly relevant.
 
9
We use the notation \(QV\) as a shorthand for the function \((QV)(w,\xi )\) defined to be \((QV)(x,\xi )\equiv \sum \nolimits _{j \in I} q_{\xi j} V(x,j)\).
 
10
The process \(Y\) is the log of the discounted asset, divided by \(\sigma \).
 
11
See [33], II.67.
 
12
See [34], VI.11.
 
13
Notice that \(\hat{\kappa }_t = \left\langle \pi _t,\kappa \right\rangle \), so that the drift in \( dY \) is expressed in terms of \(\pi _t\).
 
14
In fact, we have \(\lambda ^{\prime } = \exp (-\gamma _0 RT) w_0^{-R}\), where \(\gamma _0 = (R-1)(r+\kappa ^2/2R)/R\).
 
15
An extended account can be found in Muraviev & Rogers [29].
 
16
This is a very classical growth problem; see, for example, the book by Romer [36] for more background. We take here what is perhaps the simplest form of the problem.
 
17
We even assume that the parameters are known.
 
18
In this case, because of the time-invariance of the problem, they would in fact be choosing the same actions as the current agent.
 
19
Of course we need some conditions on \(f\); bounded Lipschitz is quite sufficient.
 
20
Recall that \(R=2>1\).
 
21
We omit consideration of the case \(R=R^{\prime }\), which is a knife-edge case.
 
22
We used the easily-verified fact that \(Q(1-1/R) = -\gamma _M\).
 
23
The assumption that \(a \le r \le b\) is merely for expositional convenience. You are invited to work out what happens if this condition does not hold.
 
24
This assumption would be correct if the Lévy process was a Brownian motion with drift, when the market is complete, but is otherwise a big ask.
 
Metadata
Title
Variations
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_2