Abstract
This chapter studies the investor’s optimization problem in an incomplete market where the investor has a utility function defined over both terminal wealth and intermediate consumption. The presentation parallels the portfolio optimization problem studied in Chap. 11, and it is based on Jarrow (Quart J Finance8:33, 2017).
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Change history
11 June 2022
The author noticed few mistakes in Chapters 3, 6, 12, 14 and 17 as shown below.
References
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Appendix
Appendix
Theorem 61 (Numeraire Invariance)
Let Zt and Yt be semimartingales under \(\mathbb {P}\).
Consider \(V_{t}=V_{0}+\int _{0}^{t}\alpha _{u}dZ_{u}\).
Then \(VY=\alpha \bullet \left (YZ\right )\) where • denotes stochastic integration.
Proof
For the notation, we use the convention in Protter [158, p. 60], related to time 0 values.
By the integration by parts formula Theorem 3 in Chap. 1 we have
V Y = Y−•V + V−•Y + [V, Y ].
For the first term we have
Y−•V = Y−•(α•Z) = (Y−α)•Z
by the associate law for stochastic integrals, Protter [158, Theorem 19, p. 62].
= (αY−)•Z = α•(Y−•Z).
For the second term,
V−•Y = (α•Z−)•Y = α•(Z−•Y ).
For the third term we have
[V, Y ] = [α•Z, Y ] = α•[Z, Y ], see Protter [158, Theorem 29, p. 75].
Combined,
V Y = α•(Y−•Z) + α•(Z−•Y ) + α•[Z, Y ].
Or, \(VY=\alpha \bullet \left ((Y_{-}\bullet Z)+(Z_{-}\bullet Y)+[Z,Y]\right )\).
But, Y Z = (Y−•Z) + (Z−•Y ) + [Z, Y ] by the integration by parts formula, Theorem 3 in Chap. 1.
Hence, \(VY=\alpha \bullet \left (YZ\right )\). This completes the proof.
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Jarrow, R.A. (2021). Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth). In: Continuous-Time Asset Pricing Theory. Springer Finance(). Springer, Cham. https://doi.org/10.1007/978-3-030-74410-6_12
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