Incomplete Markets (Utility over Intermediate Consumption and Terminal Wealth)

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. This chapter is based on Jarrow.

Appendix

Theorem 12.7 (Numeraire Invariance)

Let Z _t and Y _t 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 [151, p. 60], related to time 0 values.

By the integration by parts formula Theorem 1.1 in Chap. 1 we have

$$\displaystyle \begin{aligned} VY=Y_{-}\bullet V+V_{-}\bullet Y+[V,Y].\end{aligned} $$

For the first term we have

$$\displaystyle \begin{aligned} Y_{-}\bullet V=Y_{-}\bullet(\alpha\bullet Z)=(Y_{-}\alpha)\bullet Z\end{aligned} $$

by the associate law for stochastic integrals, Protter [151, Theorem 19, p. 62].

$$\displaystyle \begin{aligned} =(\alpha Y_{-})\bullet Z=\alpha\bullet(Y_{-}\bullet Z).\end{aligned} $$

For the second term,

$$\displaystyle \begin{aligned} V_{-}\bullet Y=(\alpha\bullet Z_{-})\bullet Y=\alpha\bullet(Z_{-}\bullet Y).\end{aligned} $$

For the third term we have

$$\displaystyle \begin{aligned}{}[V,Y]=[\alpha\bullet z,Y]=\alpha\bullet[z,Y],\end{aligned}$$

see Protter [151, Theorem 29, p. 75].

Combined,

$$\displaystyle \begin{aligned}VY=\alpha\bullet(Y_{-}\bullet z)+\alpha\bullet(z_{-}\bullet Y)+\alpha\bullet[Z,Y].\end{aligned}$$

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 1.1 in Chap. 1.

Hence, $VY=\alpha \bullet \left (YZ\right )$. This completes the proof.