# A simple decomposition for fractional Brownian motion with parameter $H<1/2$

## 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)?

Sorry I don't have time to write a better answer. I would be willing to bet Nualart has thought about this problem at least and his answer could very well be encompassed in this paper: (In particular your problem might be a special case described in section 3)

P. Lei and D. Nualart: A decomposition of the bifractional Brownian motion and some applications. Statistics and Probability Letters 79, 619-624, 2009.

http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.2227v1.pdf

• That reference is certainly interesting in our context (and I might even cite it). However, the decomposition described there for fBm (cf. Proposition 1) is not similar to the one I indicated above, and is apparently much more complicated. – Shai Covo Nov 7 '10 at 11:55

Interesting, I don't think I've seen that before. But there is a similar sort of decomposition in W. Li and W. Linde 1998, however I don't think it's quite the same. Cheridito 2003 (Mixed-FBM) tackles a tangential but not altogether unrelated question. One last comment-- you should probably put max(s,t) in your covariance function since you are implicitly assuming t>s.

It looks like this idea isn't new. There is something very similar in an article by Alos, Mazet, and Nualart (SPA 2000).

Shai Covo, have you found any references to this covariance? Alos, Mazet and Nualart, SPA 2000 use a different decomposition of fBm. Have you published your result, and if so, could you give the reference? How does the process Y arise in relation to Gaussian random fields?

I was thinking of a problem of existence of bifractional Brownian motion, and I proved that the function corresponding to your Y is nonnegative definite. I also noticed the decomposition of fBM. I searched whether this covariance appeared anywhere, and I could not find anything. My result is on arxivhttps://arxiv.org/abs/1902.09633. Recently I found your question.

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