Let $Z_{1},...,Z_{T}$ be independent standard normal random variables and let $\mathcal{F_{t}}:=\sigma(Z_{1}, ..., Z_{t})$.
Let $X_{0}^{1}>0, \sigma_{i}>0$ and $m_{i}\in \mathbb R$ be constants. The discounted price process is given by:
$X_{t}^{1}:= X_{0}^{1}\prod\limits_{i=1}^{t}e^{\sigma_{i} Z_{i} +m_{i}}$
Construct an equivalent martngale measure for $X^{1}$ under which the random variables $X_{t}^{1}$ still have a log-normal distribution:
My idea: First we need to find which conditions the Martingale Measure $Q$ needs to satisfy.
$ X_{0}^{1}\prod\limits_{i=1}^{t}e^{\sigma_{i} Z_{i} +m_{i}}=X_{t}^{1}=E[X_{t+1}^{1}\vert \mathcal{F}_{t}]=E[ X_{0}^{1}\prod\limits_{i=1}^{t+1}e^{\sigma_{i} Z_{i} +m_{i}}\vert \mathcal{F}_{t}]$
This is equivalent to:
$1= E[e^{\sigma_{t+1} Z_{t+1} +m_{t+1}}\vert \mathcal{F_{t}}]=E[e^{\sigma_{t+1} Z_{t+1} +m_{t+1}}]$
this then leads to:
$\ln(1)=E[\sigma_{t+1} Z_{t+1} +m_{t+1}]=\sigma_{t+1}E[Z_{t+1}]+m_{t+1}$
where $E$ represents the expectation operator under $Q$.
Any ideas on how to construct the Martingale Measure under $Q$ with lognormal distribution?