## 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

### Keywords:

fractional Brownian motion; Lévy theorem; martingales

Zbl 1138.60006
Full Text:

### References:

  Carlen, E. and Krée, P. (1991). L p estimates on iterated stochastic integrals. Ann. Probab. 19 354-368. · Zbl 0721.60052  Hu, Y., Nualart, D. and Song, J. (2009). Fractional martingales and characterization of the fractional Brownian motion. Ann. Probab. 37 2404-2430. · Zbl 1196.60075  Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422-437. JSTOR: · Zbl 0179.47801  Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. 1929 . Springer, Berlin. · Zbl 1138.60006  Molchan, G. M. (2002). Linear problems for fractional Brownian motion: A group approach. Teor. Veroyatnost. i Primenen. 47 59-70. · Zbl 1035.60084  Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 571-587. · Zbl 0955.60034  Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006  Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7 95-105. · Zbl 0884.90045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.