Nonlinear Functionals of Fractional Brownian Motion

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Fractional Brownian motion is neither Markov process nor semimartingale. Thus it can be and has recently been applied to describe phenomena that cannot be described by these two major stochastic processes. To enlarge the scope of application one needs to study the nonlinear functionals of fractional Brownian motion. An important and natural class of such functionals are those given by stochastic differential equations. In addition to studying self-intersection local time, the principal investigator proposes to study general stochastic differential equations driven by fractional Brownian motion. The difficulty in such study is caused by the fact that the powerful Picard's iteration approach fails to work. The principal investigator proposes to combine fractional calculus, anticipative stochastic calculus, and the characteristic theory to investigate such equations.To describe natural or social phenomena mathematically, people usually use Markov property (the future depends only on today although the whole history until today is known). This is a reasonable simplification, particularly if one considers the sophistication needed to deal with the entire past. However, it becomes more and more demanding to assimilate all the information available to better predict the future. Fractional Brownian motion is among the simplest statistical model that captures this long memory character. It has found many applications. To more adequately fit mathematical models to the phenomena under consideration, one should use fractional Brownian motion as building blocks to obtain more sophisticated random quantities. The principal investigator has focused on this statistical model for a number of years and has achieved significant success. This research will considerably further this progress and is expected to have impact on many other fields. Immediate applications are to be found in finance and bio-informatics.