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### Correlated Brownian motions across different times and representation with independent processes

Take $W^1,W^2$ to be two independent Brownian motions and let $\varrho(t)$ be a function with values in $[-1,1]$. Set  B^1_t=W^1_t\quad\text{ and }\quad B^2_t=\int_0^t\varrho(s)\,dW^1_s+\int_0^t\...
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Following the proof of Theorem 4.21, fix the environment $\mu$, and additionally suppress the dependence of the SDE coefficients on $\mu$, so that the nonlinear SDE reduces to a classical one. Claim: ...