Desarrollo de la propuesta utilizando la herramienta ESTUDIO DE MÉTODOS-

Contradictory to the normality assumption and Gaussian distributed increments that are given by the process that admits Brownian motion definition, Paul Lévy (1886-1971) has pioneered non-Gaussian processes with independent and stationary increments, which was named after him as “Lévy processes”. Lévy process is connected with the

infinitely divisible laws, through “Lévy-Khintchine formula” their distributions are characterised, and their structure are described by “Lévy-Itô decomposition”. Paul

Lévy scientific work has been gathered in (Loève, 1973).

This subsection is built on the “Brownian motion” subsection and connected to it in some parts. Previous description of the Lévy process outlines the structure of this subsection. In this subsection only the necessary definitions, theorems, and their proofs are stated. There are many good monographs that review carefully the Lévy process, for example: (Sato, 1999), (Schoutens, 2003), and (Cont and Tankov, 2004); as this subsection is mainly referred to, while for comprehensive overviews of its applications

see (Prabhu, 1998), (Barndorff-Nielsen et al., 2001), (Kyprianou et al., 2005), and (Kyprianou, 2006).

In probability theory, and as could be seen in the subsequent definition, the distribution and the density functions of a random variable are completely defined by their corresponding characteristic function. Therefore it provides the foundation of an alternative method to analytically evaluating the probability of occurrence instead of working directly with the probability distribution and density functions.

There are principally simple consequences for the characteristic functions of distributions described by the weighted sums of random variables. Additionally to univariate distribution functions of a random variable, the characteristic functions can represent vector-valued or matrix-valued random variables. Moreover, it could be extended to incorporate with more complicated and generic situations. These and some other properties will be stated below.

Definition 5.7 (Characteristic Function ")

Let Á_É(¾) and ¯_É(x) be, respectively, a distribution and density function of a random variable É those, respectively, admit Definition 3.6 and Definition 3.7, and y be the imaginary number, i.e. (Ñ = −1), then the characteristic function20 of a random variable É, denoted by z_É, is given by the subsequent equalities:

zÉ(±) = Z®(ÒBÉ) = ®(ÒBÉ)_­Á(x) ∞ '∞ = ®(ÒBx)¯É(x)­x ∞ '∞

Furthermore, the probability distribution of the random variable É’s behaviour and

properties are completely determined by its corresponding characteristic function. The

20

two methods are analogous in the sense that knowing one of them leads to find the corresponding one. However, each of them may introduce different insight for understanding the features of the random variable É.

The next corollary gives an example of how the characteristic function could be extract from its corresponding density function, i.e. applying the characteristic function, which is defined in Definition 5.7, on the Gaussian density function, which is defined in Definition 5.1, to extract the corresponding Gaussian characteristic function.

Corollary 5.1 (Gaussian Characteristic Function)

Let É_{*(k ,l)} be a Gaussian random variable that admits Definition 5.1, then its characteristic function, which admits Definition 5.7, is given by the subsequent equality: zÉ_{*(m ,o)}(±) = ®(ÒBx) 1 2{=® Ä'(x'k)_l––Å ­x ∞ '∞ = ®HÒBk'§l–B–L

Some of the characteristic function properties are articulated below. Property D5.7.1 (Characteristic Function: Existence)

For any random variable É, there exists a continuous corresponding characteristic function z_É.

When the characteristic function is considered as a function of a real-valued argument, it always exists, unlike the moment-generating function. The existence of the moment- generating and density functions depends on the behaviour and properties of the characteristic function. Therefore, the characteristic function could be seen as a strong alternative as stated above to the distribution function.

Property D5.7.2 (Characteristic Function: Independent Sequence)

characteristic function z_É, which admits Definition 5.7, is given by the subsequent equality:

zÉ(±) = zɕ(±)zɖ(±) ⋯ zÉ(±)

The independence sequence property of the characteristic function is one of its most important properties; this method is principally valuable when analysing linear combinations of independent random variables. Additionally, it has an important application when a sequence of random variables is needed to be decomposed. Furthermore, it is the base for another important concept, i.e. the “infinitely divisible distribution”.

The next three properties, i.e. cumulant function, moment generating function, cumulant characteristic function, are some related functions those correspond to the characteristic function and often appear in the literature.

Property D5.7.3 (Characteristic Function: Cumulant Function)

Let z_É be the characteristic function of a random variable É that admits Definition 5.7, then the cumulant function, denoted by |_É, is related to the characteristic function by the subsequent equality:

zÉ(y±) = ®|É(B) = ®Zà(S`É)

Property D5.7.4 (Characteristic Function: Moment Generating Function)

Let z_É be the characteristic function of a random variable É that admits Definition 5.7, then the moment generating function, denoted by }_É , is related to the characteristic function by the subsequent equality:

z_É(−y±) = ®~É(B)

= ®Zà(`É)

As stated previously, the moment-generating function does not always exist and thus the characteristic function is a better representation.

Property D5.7.5 (Characteristic Function: Cumulant Characteristic Function)

Let z_É be the characteristic function of a random variable É that admits Definition 5.7, then the cumulant characteristic function or characteristic exponent function, denoted by _É, is related to the characteristic function by the subsequent equality:

zÉ(±) = ®€É(B) = ®º Zà(‚`É)

The following property manipulates the one-to-one correspondence between the characteristic function and the density function or the distribution function. Therefore, it is possible to it is always possible to obtain one of them when the other one is known. This property is known as the characteristic inversion method or the extraction method. Property D5.7.6 (Characteristic Function: Extracting the Density Function)

Let ¯_É(x) and z_É(±) be, respectively, a density and a characteristic function of a random variable É those, respectively, admit Definition 3.7 and Definition 5.7, and y be the imaginary number, i.e. (y = −1). Then the density function of a random variable É could be extract from its corresponding characteristic function by the subsequent equality:

¯_É(x) =_2q1 ®('ÒBÉ)_z

É(±)­¾

∞

'∞

The pervious definition states that the characteristic function and its corresponding density function are paired, i.e. each of them could be seen as a Fourier Transform of the other. This depends on the existence of the density.

The characteristic function definition and this property will be the access point to the computational chapter, i.e. Chapter 8.

Definition 5.8 (Infinitely Divisible Distribution)

5.7, then z_É is said to be an infinitely divisible if and only if for every  ∈ ℕ there exsist an  power corresponding characteristic function, which could be given by the subsequent equality:

z_É(±) = ƒz_É

H_L• (±)„

The above definition intuitively means that if a random variable † could be characterised as a sum of two independent random variables with identically distributions, then † is said to be infinitely divisible.

In general to prove that a distribution is infinitely divisible, it has to be examined throughout its sequence and it has to be true on each and every point. The next corollary shows that the Gaussian distribution is an infinitely divisible.