Levy Process

Levy Process is a specific stochasitic process, named by Paul Levy who is the French mathematician. It is defined as a continuous stochastic process which has properties of independent increments and stationary increments. The typical exapmle is as we know, Brownian motion. In practice, Black Sholes formulation is often used by plain vanilla option, such as FX Option, Stock option, Swaption, and so on. Black Sholes is based on the assumption that the underlying is followed by Brownian Motion. I want to pay attention to what this process is continuous path. On the other hand, Levy process is not supposing that it is continuous path. Therefore, a Levy Process is including Brownian Motion at all. Let us think about realistic from now. Market behavior can not always be continuous. For example, when it was Black Monday or Lehman Brothers bankruptcy, the market changed in a extream range. Hence, Levy process model is more and more worth of studying in Mathematical Finance. Of course, many financial reserchars has already studied it...