I'm following Milevsky & Posner (2003) to model the value of a portfolio, $P$, assuming GBM (2) and a continuous contribution rate of $1$ (i.e. dollar-cost averaging):
$$ \mathrm d S_t/S_t = \mu \mathrm d t + \sigma \mathrm d B_t \iff S_t = S_0 \exp\left[ (\mu - \frac 1 2 \sigma^2)t + \sigma B_t \right] \tag{2} $$
$$ P_T = S_T \int_0^T \frac{\mathrm dt}{S_t} \tag{6} $$
Where $S_t$ is the stock price and $1/S_t$ is the amount of stock we can buy with $1.
$$ P_T = \int_0^T \hat S_\tau \mathrm d \tau = \int_0^T \exp\left[\mu \tau - \frac 1 2 \sigma^2 \tau + \sigma \hat B_\tau \right] \mathrm d \tau \tag{8} $$
where $\hat S_\tau \sim S_t$ and $\hat B_\tau \sim B_t$.
How can we solve this integral in terms of the normal/lognormal distribution?
How can we find, or approximate, the quintile function? I want to use this to get quartiles & 95% CI for this investment strategy.
Are there any problems with modelling the problem this way, or is it important to use more complicated models such as stochastic volatility?
I believe the solution may be:
$$ P_T = \frac 1 \mu \left( \exp \left[ \left(\mu-\frac 1 2 \sigma^2\right)T + \sigma\sqrt{T} \mathcal N \right] -1 \right) $$
where $\mathcal N \sim \text{Normal}(0,1)$ so $P_T$ is a linear function of a LogNormal dsitribution.
This gives the correct result for the expected value and the case where $\sigma = 0$, but I'm not sure how to actually derive it.
$$ \mathbb E[P_T] = P_{T\ \sigma=0} = \frac 1 \mu \left( e^{\mu T} - 1 \right) $$