[Math] Sum of two correlated geometric Brownian motions

Question

Suppose $S_1$ and $S_2$ are geometric Brownian motions satisfying
$$dS_1 = S_1[\mu_1 dt + \sigma_1 dX_1], \qquad dS_2 = S_2[\mu_2 dt + \sigma_2 dX_2],$$
where $d\langle X_1, X_2\rangle = \rho$. One can check, using Ito's Lemma for instance, that $P = (S_1+S_2)/2$ follows the process
$$dP = \frac12(dS_1 + dS_2) = \frac12(\mu_1S_1 + \mu_2S_2)dt + \frac12\sigma_1S_1 dX_1 + \frac12\sigma_2S_2 dX_2.$$
This is a bit surprising to me, since I would have thought the sum is also a GBM with the appropriate volatility and mean parameters. Furthermore, we haven't made any use of the correlation condition. So, is it true that the sum of two correlated GBMs is a GBM? What about for three correlated GBMs (with the weights summing to 1)?

Answer 1

Seems like a sum of GBMs is not a GBM.

Note that GBM has lognormal increments, so our sum of correlated GBMs must have lognormal increments if it is GBM. The answer to https://stats.stackexchange.com/questions/238529/the-sum-of-independent-lognormal-random-variables-appears-lognormal suggests that sums of lognormals are not necessarily lognormal, "not even for i.i.d. lognormals".