## Background

Let $X = \{X(t):t \geq 0\}$ be a (standard, real-valued) fractional Brownian motion (fBm) with parameter $H \in (0,1)$, i.e., a continuous centered Gaussian process with covariance function given, for $0 \leq s \leq t$, by $$C_X {(s,t)} := {\rm E}[X(s)X(t)] = \frac{1}{2}[t^{2H} + s^{2H} - (t - s)^{2H} ].$$ Writing $C_X {(s,t)}$ as $C_X {(s,t)} = \frac{1}{2}[t^{2H} - (t - s)^{2H} ] + \frac{1}{2}s^{2H}$, gives rise to the decomposition of $X$ as $X = Y + Z$, where $Y$ is a centered Gaussian process with covariance function $C_Y {(s,t)} = \frac{1}{2} [t^{2H} - (t - s)^{2H}]$, independent of a time-changed Brownian motion $Z$ (specifically, $Z(t)=W(t^{2H}/2)$, where $W$ is a standard BM). However, in order for $C_Y$ to be a valid covariance function it must be nonnegative definite. As indicated by numerical results (and can probably be easily proved), this is not the case for $H>1/2$. For $H<1/2$, on the other hand, $C_Y$ is the covariance function of some interesting Gaussian process arising in the setting of Gaussian random fields. Since I plan to write a paper on this apparently new subject, I find it sensible not to give too much details here (maybe I'll add some details later on).

Now to my questions. Have you encountered the aforementioned decomposition in the literature? (I haven't.) Does it correspond to some known (e.g., integral) representation of fBm? Can you think of some application of it? Finally, can you find a simple/useful representation for the process $Y$ in that decomposition (simple/useful compared to the fBm case)?