Stochastic Analysis and Applications

The world we live in has never been very predictable, and randomness has always been part of our lives. There is ample evidence that our ancestors did enjoy playing games of chance, and the early development of probability theory had to do with these games. The study of probability has always been motivated by potential applications. Besides gambling, the stimulus has come from life insurance, pricing of annuities, statistical modeling of errors in observations, genetics, etc. By 1900, there had developed a fairly well understood body of work, although it was not within an axiomatic framework. This prompted Hilbert to pose the axiomatization of probability theory as one of his famous problems. Although it was not until 1933 in [13] that Kolmogorov axiomatized probability theory by making it part of measure theory, the subject continued to blossom in the early part of the twentieth century at the hands of people like Lévy, Khintchin, and Wiener.