Fractional Brownian Motion Of Order Research Articles

Nonlocal fuzzy fractional stochastic evolution equations with fractional Brownian motion of order (1,2)

<abstract><p>In this manuscript, we formulate the system of fuzzy stochastic fractional evolution equations (FSFEEs) driven by fractional Brownian motion. We find the results about the existence-uniqueness of the formulated system by using the Lipschitizian conditions. By using these conditions we have also investigated the exponential stability of the solution for the above system driven by fractional Brownian motion. Finally, the applications in financial mathematics are presented and the use of financial mathematics in the fractional Black and Scholes model is also discussed. An example is propounded to show the applicability of our results.</p></abstract>

On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion

It is shown that, by using Taylor’s series of fractional order, the stochastic differential equation d x = σ x d b ( t , a ) , where b ( t , a ) is a fractional Brownian motion of order a , can be converted into an equation involving fractional derivative, therefore a solution expressed in terms of the Mittag–Leffler function.

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.

Probability density of the complex-valued fractional Brownian motion of order n via the maximum entropy principle in R+1/ n

In previous work, it has been proposed a model of complex-valued fractional Brownian motion with independent increments which is defined by means of a random walk on the complex roots of the unity. Here one shows that, exactly like the Gaussian probability function can be obtained as a result of the maximum entropy principle, the probability density of the complex-valued fractional noise of order n maximizes the informational entropy of a complex-valued random variable defined in an isotropic complex space of order n. In addition one defines and derives the main properties of the so-called coloured fractional noise of order n and its probability density also is obtained by using the maximum entropy principle. Lastly, as an illustrative example of application, one analyzes how the effects of fractal noises disturbances, in the form of fractal noises, on the stability of dynamical systems, vary depending upon of their order.

A Gaussian correlation inequality and its applications to the existence of small ball constant

Let X 1,…, X n be jointly Gaussian random variables with mean zero. It is shown that ∀ x>0 and ∀1⩽ k< n P max 1⩽i⩽n |X i|⩽x ⩽(1/ρ)P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x , P max 1⩽i⩽n |X i|⩽x ⩾ρP max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x and P max 1⩽i⩽n |X i|⩽x ⩾2 − min(k,n−k)/2 P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i |⩽x , where ρ=(| Σ|/(| Σ 11| | Σ 22|)) 1/2, Σ, Σ 11 and Σ 22 are the covariance matrices of (X 1,…,X n), (X 1,…,X k) and ( X k+1 ,…, X n ), respectively. In particular, for fractional Brownian motion { X( t), t⩾0} of order α (0< α<1), there exists d α >0 such that P sup 0⩽s⩽a |X(t)|⩽x, sup a⩽t⩽b |X(t)−X(a)|⩽y ⩾d αP sup 0⩽s⩽a |X(t)|⩽x P sup a⩽t⩽b |X(t)−X(a)|⩽y for any 0<a<b, x>0 and y>0. As an application, it is proved that the small ball constant for the fractional Brownian motion of order α exists.

Risk and Complex Fractals in Finance∶ Application to a Black-Scholes Equation of Order n

The short term risks in finance cannot be suitably analyzed by using the classical risk theory, and a way to circumvent this pitfall is to use fractional processes. The higher the risk is the higher the order of the fractal is. Recently, a model of complex-valued fractional Brownian motion of order n has been suggested in the literature, and the purpose herein is to examine which kind of results it can provide in mathematics of finance. After a short background to this model, one comments on fractals in finance, and then one suggests a general model of complex fractals for stock market. Various state variables can be identified with the parameters so involved in this model, and mainly, one can introduce a new concept of stock market time different from the standard absolute physical time. As an application, a Black-Scholes equation of order n is derived.

Randomization of transfer functions in control systems via computer vision with pixel noises of order n

Complex-valued fractional Brownian motion of order n can be defined either as rotating Gaussian white noise on the net defined by the n th roots of the unity, or as the limit of a random walk in the complex plane. After a brief background, one shows that this stochastic process is quite relevant in computer vision, as the result of the definition of image in terms of pixels. It follows that in control systems involving numerical vision, one will have to consider disturbances in the form of fractional white noise of order n with independent increments, and the present paper deals with this problem. By using the central limit theorem, it is possible to consider the Laplace transform of such a process as a fractional Gaussian variable of order n of which the n th moment is the integral of the n th moment of the white noise. One can then use this result to carry on a statistical analysis of linear feedback systems subject to disturbing fractional white noises (or pixel noises) in control via computer vision. Stochastic optimal control of order n is outlined.

A New Approach to Fractional Brownian Motion of Order n Via Random Walk in the Complex Plane

By using a very simple model of random walk defined on the roots of the unity in the complex plane, one can obtain the model of fractional brownian motion of order n which has been previously introduced in the form of rotating Gaussian white noise. This definition of fractional Brownian motion of order n as the limit of complex random walk, provides new insights in its genuine practical meaning, and in the derivation of most of the related theoretical results. Itôs stochastic calculus can be extended in a straightforward manner to the path integral so generated in the complex plane. The corresponding probability distribution is stable in Levys sense, a Lindebergs like central limit theorem is stated, together with a Feyman–Kacs formula and a Dinkins formula. Then one exhibits the relation between the Hausdorffs dimension and the pattern entropy of the process. The probabilistic approach here is different from Hochbergs and Mandelbrots. Like Saintys, it uses the complex roots of the unity, but it is much more straightforward and simple, and it is the only one which provides results which are fully consistent with the so-called Kramers–Moyal expansion.

INFORMATION AND SELF-ORGANIZATION REVISITED VIA COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION OF ORDER n

Haken obtained many interesting results on information and self-organization by using Jaynes' maximum entropy principle with constraints in the form of given values of state moments up to the fourth order. The basic contention of this approach lies on the selection of these constraints, for the very reason that the result so obtained depended heavily upon them. The mean values of the state moments are quite acceptable and understandable, but why not the fifth one? In this paper, it is shown that the use of complex valued fractional Brownian motion of order n can contribute to answering this question. In addition, one examines how Haken's results are modified when the disturbing random term is not a Gaussian white noise, but a fractional Gaussian white noise of order n. The effect of such a noise on the stability of dynamical systems is analyzed. The key is that one has to consider much more state moments than when one works with Gaussian white noise.

Construction of a complex-valued fractional Brownian motion of order N

In this paper, a Brownian motion of order n (n≳2) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent R+1/n-valued random variables (rv) (where R+1/n={z∈C; zn∈R+}). Many properties of the real standard Brownian motion are generalized at order n, but in the case n≳2, it is interesting to describe the Brownian motion of order n on the σ algebra ⊗[B(R+1/n)] R+ [where B(R+1/n) is the σ algebra generated by sets of type A(0,h)={z∈C; zn∈[0;hn[},(hεR+*)]. This σ algebra is totally different from ⊗[B(R)] R+. Thus this study shows the fractal nature of the Brownian motion of order n, and given invariance scale (self-similarity) properties. Then, a stochastic integral and an Itô–Taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average. All these results can be obtained via nonstandard analysis methods (infinitesimal time discretization). Finally, one remarks that this process has a.s (almost surely) continuous sample paths, infinite variance, and independent increments, whereas the fractional Brownian motion of Mandelbrot has a.s continuous sample paths, finite variance, and interdependent increments.