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Controlling rough paths. (English) Zbl 1058.60037
The author presents integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. The starting point is an observation that if \(f\in C(\mathbb R)\) and is bounded, \(x\in C^1(\mathbb R)\), then \(a_t=\int_0^tf_u\,dx_u\), \(r_{st}=\int_0^t(f_u-f_s)\,dx_u\) is a unique solution of the equation \(f_s(x_t-x_s)=a_t-a_s-r_{st}\), under the conditions \(\lim_{t\to s}(| r_{st}| /| t-s| )=0\), \(a\in C^1(\mathbb R)\), \(a_0=0\), \(r\in C({\mathbb R}^2)\). As this equation makes sense for any \(f, x\), it suggests a possibility to generalize \(\int f\,dx\) for functions not necessarily of finite variation. Indeed, a good notion of integral is defined with respect to paths with Hölder exponent greater than 1/3, and the problem of existence, uniqueness and continuity of solutions of the differential equation driven by such paths is studied. This generalizes Young’s theory of integration, and main results of Lyons’ theory of rough paths in Hölder topology are obtained.

MSC:
60H05 Stochastic integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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