Let $W_i(t)$ and $W_j(t)$ be correlated Brownian motions with $Corr(W_i,W_j) = p$. Suppose $S_i$ and $S_j$ are geometric Brownian motions driven by $W_i$ and $W_j$ such that: $$ dS_i = \mu_i·S_i·dt + \sigma_i·S_i·dW_i(t) \\ dS_j = \mu_j·S_j·dt + \sigma_j·S_j·dW_j(t) $$ I understand $$Corr[\log(S_i(t)/S_i(0)),\log(S_j(t)/s_j(0))] = p,$$ but could some please show that $$Corr[\log(S_i(t)/S_i(t-1)),\log(S_j(t)/S_j(t-1))] = p$$ as well? It requires computing expected value of product of Brownian motion at different times, i.e. $$\Bbb E[W_i(t)W_j(t-1)] = p*(t-1),$$ there is a previous post on this but the proof was not clear and I really hope to find this somewhere in a paper or book or a nice proof. Thank you very much!!!
2026-04-03 20:54:52.1775249692
Question on correlated geometric brownian motions (stock prices)
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