When Linear Combinations of Fractional Brownian Motions Remain Fractional Brownian

Summary

The document asks whether the familiar construction for combining correlated standard Brownian motions has an analogue for fractional Brownian motions. It starts with two independent fractional Brownian motions and considers weighting one by a correlation coefficient while weighting the other by the complementary square-root term. The central question is whether this combination is itself a fractional Brownian motion, potentially with a new Hurst parameter, when the input motions may have different Hurst parameters.

No answer or derivation is included, so the document does not establish conditions under which the resulting process has the covariance structure or other defining properties of fractional Brownian motion. It is best read as a modeling question about combining processes with different dependence and scaling behavior, not as a ready-to-use construction. The discussion also leaves open what “correlated” should mean for the full processes, beyond the coefficient in the proposed linear combination.

Key ideas

• The document asks whether independent fractional Brownian motions can be combined by a weighted sum.
• It considers input motions that may have different Hurst parameters.
• A valid construction would need to preserve the defining properties of fractional Brownian motion.
• The document provides no solution or conditions for when such a combination works.

Tags

# mixing fractional Brownian motions


# mixing fractional Brownian motions












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.