## More Math: the Lévy Distribution

The Lévy distribution is a probability distribution. Here it is expressed as a Fourier transform, ‘F’, of the probability function PN(k) for random variables with N steps. On the right-hand side of the equation, β (the Greek letter “beta”) is a parameter. Lévy had proven that β must be in the range from zero to two for the probability P(x) to be non-negative. Setting β to 1 in the equation generates the Cauchy distribution; while the normal distribution comes from β = 2. The Lévy distribution has an infinite variance. Although it may predict “average” values, the outliers can range very far afield.

## The T Cell Research

A “T Cell” is a type of leukocyte, or colourless blood cell – the formal name for these cells is “T Lymphocyte.” The ‘T’ indicates that the thymus gland is involved in its activation. Unlike the granular leukocytes and monocytes which engulf foreign material, T Lymphocyte cells serve a variety of roles in regulating cellular immune responses. In this image, the T cell is at the far right.

## Lévy Processes Apply to Financial Modeling

In addition to the news item from biology, academics recommend using mathematical processes based on the Lévy Flight to model financial returns as well. Often these probabilistic analyses are based on the normal distribution, with a broader centre for the “bell curve” and a very low “tail.” A Lévy distribution, by contrast, has a narrow central peak, but a taller “tail.” The advantage is that the Lévy Walk can model rare but significant events, as well as frequent insignificant changes.

Consider the price and yield on a corporate bond. Over a period of weeks, small fluctuations in interest rates, currency exchange rates, or even the stock values will create minor changes in the bond yield. It is possible, however, that the company may file for bankruptcy; this would have a huge impact on the bond’s price. There is a parallel to the hunting strategy described above: search within a small area for some time, but move much farther away on occasion to find a new hunting ground. Also, the Lévy distribution has parameters that can be fitted, or tuned, to simulate a variety of conditions. It is, therefore, useful for modeling situations where, for example, the expected yield on a corporate bond varies slightly as interest rates change.

## Who was Paul Pierre Lévy?

Paul Pierre Lévy was born in Paris, France in 1886 to a family of mathematicians. After spending 1906 in military service, he resumed his studies and received his *Docteur ès Sciences *degree in 1912. Lévy’s work included functional analysis, the calculus of probabilities including Gaussian analysis, partial differential equations and Laplace transforms. Lévy died in Paris in 1971, after making significant contributions to changing the study of probability theory from a hodgepodge of isolated problems into a rapidly maturing branch of mathematics.

## Completing Our Stroll through the Lévy Walk

The Lévy random walk is based on the Lévy distribution, and is related to the bell curve described by the normal distribution. It aptly describes the search patterns of predators hunting hidden game, as well as some aspects of financial modeling, and the behavior of T cells, according to a new study by the University of Pennsylvania. Each of these applications shelters under the mathematics umbrella of probability theory.

**References**:

University of Pennsylvania. *T Cells ‘Hunt’ Parasites Like Animal Predators Seek Prey, a New Study Reveals*. (2012). Accessed June 3, 2012.

Dorland’s Medical Dictionary for Health Consumers. *Leukocyte* and *Lymphocyte*. (2007). Accessed June 3, 2012.

O’Connor, J., MER. *Paul Pierre Lévy*. (2000). University of St. Andrews, Scotland. Accessed June 3, 2012.

Weisstein, Eric W. *Lévy Distribution, Lévy Flight *and* Random Walk*. (2012). MathWorld-A Wolfram Web Resource. Accessed June 3, 2012.

Wu, Liuren. *Modeling Financial Security Returns Using Lévy Processes*. (2006). City University of New York. Accessed June 3, 2012.

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