If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested?
For example, modelling a stock as GBM without contributions, $ \mathrm d S = \mu S \mathrm d t + \sigma S \mathrm d B $, gives us a lognormal distribution, $\mathrm{Lognormal(\mu t, \sigma^2t)}$. If we add contributions, $ \mathrm d S = (\mu S + C) \mathrm d t + \sigma S \mathrm d B $, what do we get?
Milevsky & Posner (2003) derive the following:
$$ \mathrm d S_t/S_t = \mu \mathrm d t + \sigma \mathrm d B_t \iff S_t = S_0 \exp\left[ (\mu - \frac 1 2 \sigma^2)t + \sigma B_t \right] \tag{2} $$
$$ P_T = C S_T \int_0^T \frac{\mathrm dt}{S_t} \tag{6} $$
Where $S_t$ is the stock price and $1/S_t$ is the amount of stock we can buy with $1.
$$ P_T = C\int_0^T \hat S_\tau \mathrm d \tau = C\int_0^T \exp\left[\mu \tau - \frac 1 2 \sigma^2 \tau + \sigma \hat B_\tau \right] \mathrm d \tau \tag{8} $$
where $\hat S_\tau \sim S_t$ and $\hat B_\tau \sim B_t$.
I understand (6), but how do we derive it from the differential equation?
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?
