| The classical central limit theorem states that a sufficiently large number of identically distributed independent random variables each with finite mean and variance will be approximately normally distributed. In attempts to generalize the central limit theorem to random variables without second moments, yet maintaining self-similarity properties of the summands, Paul Levy introduced alpha stable random variables and distributions. Many real world phenomena, such as financial data, display interesting heavy-tailed distributions which are obviously non-Gaussian. In fact, Benoit Mandelbrot proposed that cotton prices follow a skewed alpha stable distribution with alpha equal to 1.7. Therefore, a detailed study of alpha stable random variables and distributions is necessary to model many economic and physical problems. The theories of classical integration and differential equations focus on deterministic integrators. Consequently, they can not handle as integrators random noise that might fluctuate rapidly over a period of time such as the highly irregular sample paths of Brownian motion. It, proposed a novel method to define an integral with respect to Brownian motion, giving rise to the new field of stochastic calculus with respect to Brownian motion. Subsequently, the theories of stochastic integration and stochastic differential equations have found important applications in a variety of disciplines including finance, biology, physics, etc. Since alpha stable random variables and alpha stable Levy motion are natural heavy-tailed generalizations of Gaussian random variables and Brownian motion, they attract lots of interest from researchers. Unfortunately, not all stochastic differential equations have analytic solutions. Therefore, numerical solution of stochastic differential equations has become very important. |
