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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Notes
- 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.
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Rogers, L.C.G. (2013). Variations. In: Optimal Investment. SpringerBriefs in Quantitative Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35202-7_2
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DOI: https://doi.org/10.1007/978-3-642-35202-7_2
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