From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics
- Magnen, J.
- Unterberger, J.
DOIAbstract
86 pages, 5 figuresInternational audienceLet $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of constructive field theory, in particular, cluster expansion and renormalization. The construction extends to a l...
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