Assume there are infinite geometric Brownian motions $S_{i}:i\in\left[0,1\right]$ where $dS_{i}=S_{i}\left[\mu_{i}dt+\sigma_{i}dW_{i}\left(t\right)\right]$ where $d\left\langle W_{i}\left(t\right),W_{j}\left(t\right)\right\rangle =\rho_{ij}dt$. I am now considering a problem: in what conditions we have $\int_{0}^{1}x_{i}S_{i}di$ is a geometric Brownian motion? Do anyone have an idea?
2026-04-29 08:42:56.1777452176
Can the integral of infinite geometric Brownian motions is a geometric Brownian motion?
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