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Rough path analysis via fractional calculus. (English) Zbl 1175.60061
In the present paper, the authors analyse the dynamical system $dx_t=f(x_t) dy_t,\;x(0)=x_0,\tag{1}$ where $$x$$ and $$y$$ are Hölder continuous of order $$\beta\in(\tfrac 13,\tfrac 12)$$. First, the authors define the integrals of the form $\int^b_af(x_t)dy_t$ using fractional calculus and obtain an explicit formula for this integral. Secondly, they prove the existence, uniqueness and continuous dependence of a solution for (1) in the input parameters $$x_0,y$$ and $$y\otimes y$$. Finally, the authors discuss the applications of these results to stochastic integrals and stochastic differential equations.

##### MSC:
 60L20 Rough paths 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H05 Stochastic integrals 26A42 Integrals of Riemann, Stieltjes and Lebesgue type 26A33 Fractional derivatives and integrals 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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