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Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given mean and variance and sum of all $x_i$ is constrained to be $1.$ I don't know much about this topic so any references would be really useful.

Additionally can mean and variance of each variable be selected independently of each other.

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  • $\begingroup$ In "Diffusions on the Simplex from Brownian Motions on Hypersurfaces" they obtain explicit SDE systems for BMs in simplices. So I suppose one can then start from simulating those. $\endgroup$
    – Thomas Kojar
    Commented Oct 10, 2023 at 1:33
  • $\begingroup$ I have solved the problem approximately i think the reference above is kind of too complicated there is a simpler solution possible i will post it after i have perfected it or published it somewhere $\endgroup$
    – arrhhh
    Commented Oct 13, 2023 at 15:09

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