Gubinelli Controlled Paths, the Sewing Lemma, and the Rough Integral

Job: develop the controlled paths (Gubinelli), the sewing lemma, the rough integral / Itô–Lyons map, and the RDE existence — with the rough integral / Itô–Lyons continuity as the worked example.

Core

The controlled paths (Gubinelli)

Controlled / Gubinelli derivative / Taylor Description
Controlled \(Y\) \(Y_{s,t}=Y'_s X_{s,t}+R_{s,t}\), $
Gubinelli derivative \(Y'\) the coefficient (the Taylor-like, the derivative of \(Y\) w.r.t. \(X\))
Taylor the \(Y\approx Y'X+R\) the Taylor expansion (the controlled the analog)

The sewing lemma

Sewing / \(\Xi\) / \(\zeta\) / rough integral Description
Sewing lemma \(\Xi_{s,t}\) the approximate increment, $
Rough integral \(\int Y\,d\mathbf{X}=\lim\sum(Y_{t_i}X_{t_i,t_{i+1}}+Y'_{t_i}\mathbb{X}_{t_i,t_{i+1}})\) (the controlled + the sewing)

The Itô–Lyons map and the RDE

Itô–Lyons / RDE / Davie Description
Itô–Lyons map \(\mathbf{X}\mapsto Y\) continuous (the Lyons, the \(d_{p\text{-var}}\) the rough metric)
RDE \(dY=f(Y)\,d\mathbf{X}\), \(f\in C^{[p]+1}\) (the existence / uniqueness)
Davie the RDE the expansion (the Davie, the Taylor-like the RDE)

See it

Rough integral \(\int Y\,d\mathbf{X}\) via the controlled / sewing and the Itô–Lyons continuity. The \(Y=Y'X+R\) (the controlled, the \(Y'\) the Gubinelli derivative), the \(\int Y\,d\mathbf{X}=\sum(Y_{t_i}X_{t_i,t_{i+1}}+Y'_{t_i}\mathbb{X}_{t_i,t_{i+1}})+O\) (the rough integral, the \(Y X+Y'\mathbb{X}\)). The Itô–Lyons: the \(\mathbf{X}\mapsto Y\) the continuous (the \(d_{p\text{-var}}\) the rough metric, the Lyons). The key point: the rough integral the controlled / sewing, the Itô–Lyons the continuous — the rough path the pathwise.

Check

Show that the sewing lemma gives the rough integral, and explain the Itô–Lyons continuity.

The sewing: the \(\Xi_{s,t}=Y_s X_{s,t}+Y'_s\mathbb{X}_{s,t}\) (the approximate, the \(|\Xi_{s,t}-\Xi_{s,u}-\Xi_{u,t}|\le\varepsilon\) the Chen / the controlled), the \(\zeta=\int Y\,d\mathbf{X}\) (the sewing, the \(|\zeta-\Xi|\le C\varepsilon\)). The Itô–Lyons: the \(\mathbf{X}\mapsto Y\) (the RDE \(dY=f(Y)\,d\mathbf{X}\)), the continuous (the \(d_{p\text{-var}}(\mathbf{X},\tilde{\mathbf{X}})\to0\) \(\Rightarrow\) \(d(Y,\tilde Y)\to0\), the Lyons). So the sewing the rough integral, the Itô–Lyons the continuous.

Explain the Gubinelli controlled path and the derivative, and the Taylor analogy.

The controlled: \(Y_{s,t}=Y'_s X_{s,t}+R_{s,t}\) (the \(Y'\) the Gubinelli derivative, the \(R\) the remainder \(O(|t-s|^{2\alpha})\)). The Taylor: the \(Y\approx Y'X+R\) the Taylor (the \(Y'\) the first-order, the \(R\) the second-order, the analog of the Taylor expansion). The rough integral: the \(\int Y\,d\mathbf{X}=Y X+Y'\mathbb{X}+O\) (the \(Y'\) the coefficient, the \(\mathbb{X}\) the second-order the iterated integral). So the controlled the Taylor, the \(Y'\) the derivative.

Pitfalls

  • The sewing lemma the \(\Xi\) the approximate, the \(\zeta\) the exact (the \(|\zeta-\Xi|\le C\varepsilon\)) — the sewing the integral, the \(\varepsilon\) the error.
  • The Itô–Lyons the continuous (the \(\mathbf{X}\mapsto Y\), the rough metric) — the Itô–Lyons the stability, the Itô the not-continuous (the uniform).
  • The RDE the existence (\(f\in C^{[p]+1}\), the existence / uniqueness) — the RDE the \(f\) the smooth, the SPDE the generalization.
  • The Davie (the RDE the expansion, the Taylor-like) — the Davie the expansion, the RDE the Taylor.

Refs

  • Gubinelli, "Controlling rough paths" (JFA 2004); the controlled paths
  • Lyons, "Differential equations driven by rough signals" (RMI 1998); the Itô–Lyons map
  • Friz–Hairer, A Course on Rough Paths (2nd ed.); the sewing lemma
  • Davie, "Differential equations driven by rough paths" (2008); the RDE expansion

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