System Control and Rough Paths

From random walks to rough paths

“…This closure is a Polish space for the metric -Höl and is denoted by 0, -Höl where ∘ denotes the Stratonovich differential of . The resulting "generalized" ODE solution driven by B can then be identified as the classical Stratonovich SDE solution [7,9,10]. This provides an essentially deterministic approach to SDE theory with numerous benefits when it comes to regularity questions of the Itô map, construction of stochastic flows, etc.…”

Large deviations for rough paths of the fractional Brownian motion

“…His approach provides a kind of pathwise calculus well-suited for system control in a stochastic context. We refer the reader to [14] and [12], where the basic ingredients of the theory are presented.…”

“…They shall be denoted by D p (R d ). Indeed, linear interpolations of interesting examples like Brownian motion, B-valued Wiener process, free Brownian motion and fractional Brownian motion have been successfully used to define the corresponding geometric rough path (see [14], [10], [3], [4], respectively). …”

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

“…The corresponding Stratonovich integral, obtained as a limit either by linear interpolation or by more refined Gaussian approximations [11,51,58,59], has been shown to diverge as soon as α ≤ 1/4. This seemingly no-go theorem, although clear and derived by straightforward computations that we reproduce in short in section 1, appears to be a puzzle when put in front of the results of rough path theory [43,44,28,39,40,20]. The essential idea conveyed by this theory -we shall make this precise in section 2 -is that a path Γ : R → R d with Hölder regularity index α ∈ (0, 1) must be seen as the projection onto the d first components of some "essentially arbitrary" rough path over Γ, denoted by …”