Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other.

My question then: is there any similar relationship between fractional Brownian motions? In other words, given $W_t^{H_2}, Z_t^{H_3}$ two independent fractional Bm, can we say anything about $$\rho W_t^{H_2}+\sqrt{1-\rho^2}Z_t$$ or $$\rho W_t^{H_2}+\sqrt{1-\rho^2}Z_t^{H_3}$$ for $H_2$ and $H_3$ not necessarily equal? Can they generate a new correlated fBm $W_t^{H_1}$?


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