Suppose that a security price $P$ follows a geometric Brownian motion $($GBM$)$ under the real$-$world measure:

$\frac{dP(t)}{P(t)}=dt+dz,$

where $P(0)=1$ and $z(T)$ is a normal random variable under this measure, with the probability density:

$f(z(T))=\frac{1}{\sqrt[\phantom{2}]{2T}}{e}^{-\frac{z(T{)}^2}{2T}}$

Under the probability measure $Q$ :

$f^Q(z(T))=f(z(T))Q(z(T)).$

where:

$Q(T)=e^{-\frac{{}^2}{2}T+z(T)}.$

Note that the expectation of any random variable $x(T)$ under the measure $Q$ is:

$E^Q[x(T)]=E[Q(T)x(T)]$

$($a$)$ Show that:

$f^Q(z(T))=\frac{1}{\sqrt[\phantom{2}]{2T}}{e}^{-\frac{z^Q(T{)}^2}{2T}},$

where

$z^Q(T)=z(T)-T.$

$($b$)$ Show that under $Q$ the security price $P$ follows the GBM:

$\frac{dP(t)}{P(t)}=(+{}^2)dt+d{z}^Q.$

$($c$)$ Now, assume a discrete$-$time stochastic discount factor $($SDF$)$ in

$M_{t+1}=e^{-y_t-\frac{1}{2}{}^2{}^2+u_{t+1}},$

where $u_{t+1}N(0,{}^2).$ Decompose the SDF into a time$-$discount component and a martingale component $Q_{t+1},$ different from that used in points $($a$)$ and $($b$).$ Show that $Q_{t+1}$ is indeed a martingale by showing that

$E_t[Q_{t+T}]=1$

for any $T>t.[$Hint: Use the low of iterated expectations:

$($d$)$ Use the martingale you found in $($c$)$ to change the SDF pricing equation under the real$-$world measure

$P_t=E_t[M_{t+1}{x}_{t+1}]$

to the pricing equation under the $Q$ measure. Give the interpretation to the measure $Q.[$Hint: Use equation $(16).]$