[Math] Law of Iterated Logarithm for Fractional Brownian Motion

brownian motionprobability theoryreference-requeststochastic-processes

(Cross-posted to https://mathoverflow.net/questions/281342/reference-for-lil-for-fractional-brownian-motion.)

Let $(\Omega,\mathcal F,\Bbb P)$ be a complete probability space.

If $B=(B_t)_{t\ge0}$ is a standard real valued Brownian Motion on this space, then the following things are well known:

  1. The trajectories of $B$ are a.s. $\alpha$-Holder continuous for all $0<\alpha<1/2$.
  2. $B\notin\operatorname{BV}[0,T]$ but a.s. $B$ has finite quadratic variation, and if I remember well this is the best possible, in sense that a.s. $\|B\|_p=+\infty$ for every $p<2$ (here $\|\cdot\|_p$ is the $p$-variation norm on the fixed interval $[0,T]$).
  3. A.s. the law of iterated logarithm says that
    $$ \limsup_{t\to0^+}\frac{B_t}{\sqrt{2t\log\log(1/t)}}=1$$
    $$ \liminf_{t\to0^+}\frac{B_t}{\sqrt{2t\log\log(1/t)}}=-1$$
    4.Set $S_t:=\sup_{0\le s\le t}B_s$ and fixed $a>0$, then reflection principle ensures that
    $$
    \Bbb P(S_t\ge a)=\Bbb P(|B_t|\ge a)
    $$
    from which we get the distribution of $S$.

My questions are: what can we say about the Fractional Brownian Motion $B^H$ of Hurst parameter $0<H<1$ (I'm particularly interested in the case of $1/2<H<1$)? And what about $|B^H|$?

  1. It is well known that a.s. $B^H$ has trajectories which are $\alpha$-Holder continuous for all $0<\alpha<H$; the same holds for $|B^H|$ since the composition of an $\alpha$-Holder and a $\beta$-Holder function is $\alpha\beta$-Holder continous.
  2. I'm not sure, but it seems reasonable that $\|B^H\|_{1/H}<+\infty$ and $\|B^H\|_p=+\infty$ for every $p<1/H$.

What about $|B^H|$?

For the third property, I don't know neither how to conjecture an analogous claim.

  1. Is it true that, putting $S_t^H:=\sup_{0\le s\le t}B_t^H$
    $$
    \Bbb P(S_t^H\ge a)=\Bbb P(|B_t^H|\ge a)\;?
    $$

What about $|S_t^H|:=\sup_{0\le s\le t}|B_t^H|$.

I bet all these properties are collected somewhere, in some lecture note, so any appropriate reference can be a perfect answer!

Best Answer

  1. The exact modulus of continuity for fBm is $\delta^{H} |\log \delta|^{1/2}$, see e.g. Lifshitz Gaussian Random Functions, p. 220. In particular, it is $\alpha$-Hölder for any $\alpha\in (0,H)$.

  2. This is very well known and is written in many introductory texts about fBm, see e.g. Nourdin Selected Aspects of Fractional Brownian Motion, Corollary 2.1.

  3. Yes, there is a law of iterated logarithm $$ \limsup_{t\to 0+} \frac{B_t^H}{t^H|2\log\log t|^{1/2}} = 1 $$ almost surely, see here, p. 36.

  4. No, neither of this is true. The distribution of maximum of fBm (or its absolute value) is quite involved.

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