## 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:

 [1] Carlen, E. and Krée, P. (1991). L p estimates on iterated stochastic integrals. Ann. Probab. 19 354-368. · Zbl 0721.60052 [2] 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 [3] 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 [4] Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. 1929 . Springer, Berlin. · Zbl 1138.60006 [5] Molchan, G. M. (2002). Linear problems for fractional Brownian motion: A group approach. Teor. Veroyatnost. i Primenen. 47 59-70. · Zbl 1035.60084 [6] 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 [7] 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 [8] Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7 95-105. · Zbl 0884.90045
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