New algebraic and physical approaches of fractional stochastic calculus
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Mon, 18/10/2010 14:15 |
Jeremie Unterberger |
Stochastic Analysis Seminar  |
Eagle House |
| Rough path theory, invented by T. Lyons, is a successful and general method for solving ordinary or stochastic differential equations driven by irregular Hölder paths, relying on the definition of a finite number of substitutes of iterated integrals satisfying definite algebraic and regularity properties. Although these are known to exist, many questions are still open, in particular: (1) "how many" possible choices are there ? (2) how to construct one explicitly ? (3) what is the connection to "true" iterated integrals obtained by an approximation scheme ? In a series of papers, we (1) showed that "formal" rough paths (leaving aside regularity) were exactly determined by so-called "tree data"; (2) gave several explicit constructions, the most recent ones relying on quantum field renormalization methods; (3) obtained with J. Magnen (Laboratoire de Physique Theorique, Ecole Polytechnique) a Lévy area for fractional Brownian motion with Hurst index <1/4 as the limit in law of iterated integrals of a non-Gaussian interacting process, thus calling for a redefinition of the process itself. The latter construction belongs to the field of high energy physics, and as such established by using constructive field theory and renormalization; it should extend to a general rough path (work in progress). | |||