I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, $\sigma_i$ for i = { 1, 2, ..., n} participants. I want to calculate for each person the probability that they might win given their $\mu_i$, $\sigma_i$ estimated from their own pattern of points accumulation in the past. I believe the expected total points at time $T$ for participant $i$ is E[S$_i$$_t$] = S$_0$$e^{x\mu_it}$ and the variance is S$_0^2$$e^{2x\mu_it}$($e^{\sigma_i^2t} -1$).
My questions are first, does the formulation seem correct for the nature of the competition? Second, how can I calculate the probability that participant $i$ will win? Bonus points if the proposed solution can be done with spreadsheet formulas in closed form (no simulation) for $n$ participants.
I performed a quick simulation study with 3 participants over 1 time step (picture below) with them having the following parameters: $$\mu_1 = 2, \sigma_1 = 1$$ $$\mu_2 = 3, \sigma_2 = 1$$ $$\mu_3 = 4, \sigma_3 = 1$$
The simulation seems to suggest that Person 1 wins 4.6%, Person 2 wins 22.6% and Person 3 wins 72.6%. And I've mapped out what I think is their expected points trajectory over time step 1.
