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An extension of the Lévy characterization to fractional Brownian motion. (English) Zbl 1227.60051

Let \(X\) be a continuous square integrable zero mean random process on some probability space. The main theorem states that for a continuous square integrable centered random process \(X\), the following properties are equivalent:
(i)
the process \(X\) is a fractional Brownian motion with self-similarity index \(H\in(0,1)\); and
(ii)
the process \(X\) has the properties (a), (b), (c) for some \(H\in[0,1]\).
These properties are as follows:
(a)
the sample paths of the process \(X\) are \(\beta\)-Hölder continuous for any \(\beta\in(0,H)\);
(b)
for \(t>0\), \[ n^{2H-1}\sum^{n}\limits_{k=1}\left(X_{t_k}-X_{t_{k-1}}\right)\overset{L^{1}(p)}{\longrightarrow }t^{2H} \quad \text{as} \quad n\rightarrow\infty; \]
(c)
the process \[ M_t=\int^{t}\limits_{0}s^{\frac{1}{2}-H}(t-s)^{\frac{1}{2}-H}dX_s \]
is a martingale with respect to the filtration \(F^X\).
This theorem extends the classical characterization theorem due to P. Lévy for the Brownian motion \((H=\frac{1}{2})\) to the fractional Brownian motion.
Section 2 contains one auxiliary result for orthogonal martingales. In Sections 3 and 4, the theorem is proved for \(H>\frac{1}{2}\) and \(H<\frac{1}{2}\).
In order to prove that \(X\) is a special Gaussian process, the representation of the process \(X\) with respect to a certain martingale is applied. A different proof of the theorem appeared in Y. Mishura [Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Mathematics 1929. Berlin: Springer (2008; Zbl 1138.60006)].

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60H99 Stochastic analysis

Citations:

Zbl 1138.60006
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References:

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