# Sum of two correlated geometric Brownian motions

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)?

• Any insights, anyone? – user369210 Mar 16 '18 at 22:39