Mathematical Sciences: Stochastic Integration and Differential Equations

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The principal investigator will work on two projects in collaboration with many US and foreign scientist. Both projects relate to stochastic integrals and stochastic differential equations. The many types of stochastic integrals that exist at the moment have developed on an ad hoc basis, to fit the particular needs of the situation. The principal investigator will develop a general unifying theory based on semi-martingale integrators. It is hoped that in the process, the traditional assumption of asymptotic quasi-left continuity can be dropped. This will make the theory much more applicable. A particular application of interest is in finance theory for the extension of the Black-Scholes model. The second part of the project will deal with defining stochastic integrals with anticipating integrands. In traditional definition of stochastic integrals, the integrand (non-anticipating) is assumed to be dependent only on the past of the process with respect to which the integral is carried out. This makes the convergence properties of the integrals easy to prove. However, in many applications, such as in Volterra equations in Physics, the integrand cannot be assumed to be non- anticipating. The work will use many technical results that the principal investigator has developed in the past.