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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?

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    $\begingroup$ You may want to read the literature on DCA in c.t. such as Milevsky and Posner (2003) or Constatinides (1979). Which I will admit I have not looked at closely. $\endgroup$
    – nbbo2
    Oct 9, 2021 at 10:58
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    $\begingroup$ Instead of looking at the price distribution of your dollar cost averaged portfolio, it might actually be easier (and perhaps more useful) to look at the return distribution of the dollar cost averaged portfolio. $\endgroup$
    – user34971
    Oct 9, 2021 at 11:07

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