Rough Paths: Foundations¶

Job: develop controlled ODEs via $p$-variation and signatures; define Lyons rough paths; compare Itô calculus; point to regularity structures.

Core¶

Controlled ODEs and Young integrals¶

Consider $$ dY_t=f(Y_t)\,dX_t,\qquad Y_0=y_0, $$ with $X:[0,T]\to V$ (Banach), $f:W\to L(V,W)$ sufficiently smooth, $Y$ $W$-valued.

Riemann–Stieltjes works for $X$ of bounded variation. Young (1936): if $X\in C^{p\text{-var}}$, $Y\in C^{q\text{-var}}$ and $1/p+1/q>1$, then $\int Y\,dX$ exists as a limit of RS sums and $$ \Bigl|\int_s^t Y_{s,u}\,dX_u\Bigr|\lesssim |Y|{q\text{-var};[s,t]}|X|. $$ };[s,t]Sewing lemma (modern package) abstracts existence of integrals from approximate two-parameter increments.

ODE in Young regime. If $X$ has $p$-variation $p<2$ and $f\in C^{1+\varepsilon}$, unique solution $Y$ of finite $p$-variation exists and the Itô–Lyons map $X\mapsto Y$ is continuous in $p$-variation topology.

$p$-variation¶

For a path $X$, $$ |X|{p\text{-var};[0,T]}^p=\sup|^p. $$ Hölder-}}\sum_i |X_{t_{i+1}}-X_{t_i$\alpha$ paths have finite $p$-variation for $p>1/\alpha$. Brownian motion: finite $p$-variation a.s.\ for every $p>2$, infinite for $p\le 2$.

Regime	Tool
$p\in[1,2)$	Young integration / ODE
$p\in[2,3)$	level-$2$ rough paths
$p\in[3,4)$	level-$3$ (need triple iterated integrals)

Signature¶

For smooth $X$, the signature is the tensor series $$ \mathscr{S}(X){s,t}=\Bigl(1,\,X,\,\ldots\Bigr)\in T((V)). $$ },\,\int_s^t X_{s,u}\otimes dX_u,\,\int!!\int_{s<u_1<u_2<t}dX_{u_1}\otimes dX_{u_2Chen’s identity: $\mathscr{S}(X)_{s,t}\otimes\mathscr{S}(X)_{t,u}=\mathscr{S}(X)_{s,u}$ (multiplicative functional). Signature is invariant under tree-like reparameterized excursions; determines the path modulo tree-like equivalence (Hambly–Lyons).

Truncated signatures live in the step-$n$ free nilpotent group $G^n(V)\subset T^{(n)}(V)$ — geometric rough paths are $G^{[p]}(V)$-valued paths with Hölder/variation regularity.

Lyons rough paths¶

A rough path of roughness $p\in[2,3)$ over $V$ is a pair $\mathbf{X}=(X,\mathbb{X})$ with $$ X\in C^{p\text{-var}}([0,T],V),\qquad \mathbb{X}\in C^{p/2\text{-var}}([0,T]^2_{\le},V\otimes V), $$ satisfying Chen’s relation $$ \mathbb{X}{s,u}=\mathbb{X}}+\mathbb{X{t,u}+X $$ and analytic bounds }\otimes X_{t,u$|\mathbb{X}_{s,t}|\lesssim|t-s|^{2/p}$. (Hölder formulation: replace $p$-var by $\alpha$-Hölder, $\alpha=1/p\in(1/3,1/2]$.)

Geometric rough paths: $\mathbb{X}$ takes values so that $(X,\mathbb{X})$ lies in the group-like elements (limits of smooth path signatures). Lyons lift: smooth paths embed; rough paths are limits in a rough metric $d_{p\text{-var}}$.

Controlled paths (Gubinelli). $Y$ is controlled by $X$ if $$ Y_{s,t}=Y's X), $$ with remainder of higher Hölder regularity. Then }+R_{s,t},\qquad |R_{s,t}|=O(|t-s|^{2\alpha$\int Y\,d\mathbf{X}$ is defined by compensating with $\mathbb{X}$: $$ \int_s^t Y\,d\mathbf{X}:=\lim\sum \bigl(Y_{t_i}X_{t_i,t_{i+1}}+Y'{t_i}\mathbb{X}\bigr). $$}

Lyons’ continuity / universal limit theorem. The solution map $\mathbf{X}\mapsto Y$ for $dY=f(Y)\,d\mathbf{X}$ is continuous from rough path space into path space (for $f\in C^{[p]+1}$). Stability of ODEs under rough enrichment.

Itô vs rough¶

Feature	Itô / SDE	Rough path RDE
Integral	probability (filtration)	pathwise from $(X,\mathbb{X})$
BM enhancement	quadratic variation fixed	choose Itô or Stratonovich $\mathbb{B}$
Continuity in driver	fails in uniform norm on $B$	holds in rough metric
Chain rule	Itô correction	geometric $\Leftrightarrow$ ordinary chain rule
Non-semimartingales	limited	fractional BM, etc., if liftable

Itô–Stratonovich conversion is a change of second level $\mathbb{B}$ by a multiple of $t\cdot\mathrm{Sym}$ (trace terms) — the rough path sees the conversion explicitly.

Link to regularity structures¶

Rough paths handle paths (1D time) valued in finite-dimensional $V$. Regularity structures (Hairer) generalize the expansion philosophy to multi-parameter / distribution-valued drivers: models encode renormalized multilinear functionals of noise (higher “branched” iterated integrals); modelled distributions are the analogue of controlled paths. RDEs are the simplest regularity structure problems. See SPDE foundations.

Branched rough paths (Gubinelli) sit between classical geometric rough paths and full regularity structures when the nonlinearity needs Hopf-algebraic trees beyond tensor algebra.

See it¶

Fractional Brownian motion. For Hurst $H>1/4$, fBM admits a canonical geometric rough path lift (with $p>1/H$); for $H>1/3$ level-$2$ suffices. RDEs driven by fBM are pathwise well-posed without semimartingale structure — a result outside classical Itô theory.

Signature methods in ML / statistics. Truncated signatures of time series as feature maps: universal nonlinearity (Stone–Weierstrass on path space modulo tree-like), used in quantitative finance and sequential data. Factorial decay of expected signatures of BM gives quantitative approximation.

Check¶

Derive Chen’s relation for the level-$2$ iterated integral of a smooth path.

$\int_s^u X_{s,r}\otimes dX_r=\int_s^t+\int_t^u$. Split $\int_t^u X_{s,r}\otimes dX_r=\int_t^u(X_{s,t}+X_{t,r})\otimes dX_r=X_{s,t}\otimes X_{t,u}+\int_t^u X_{t,r}\otimes dX_r$. Rearrange to get $\mathbb{X}_{s,u}=\mathbb{X}_{s,t}+\mathbb{X}_{t,u}+X_{s,t}\otimes X_{t,u}$.

Why do geometric rough paths obey the ordinary chain rule for $Z=\phi(Y)$ along an RDE?

Geometric rough paths are limits of smooth drivers; for smooth $X$, ODE chain rule holds exactly. Continuity of the Itô–Lyons map passes the identity $\mathrm{d}\phi(Y)=\mathrm{D}\phi(Y)\,\mathrm{d}Y$ to the limit — no Itô second-order correction. Non-geometric lifts retain correction terms.

Pitfalls¶

Second level is not determined by $X$ in general: many rough paths share the same $X$ (e.g.\ Itô vs Stratonovich).
$p=2$ boundary: BM needs $p>2$; logarithmic refinements exist but standard Young still fails.
Dimension of tensor algebra: $\dim V^{\otimes k}$ grows fast — computational signature methods need low-rank / expected signature tricks.
Banach infinite-dimensional $V$: extra UMD / type conditions sometimes appear; finite-dim statements do not always copy verbatim.

Refs¶

Lyons, Rev. Mat. Iberoamericana 1998
Friz–Hairer, A Course on Rough Paths (2nd ed.)
Friz–Victoir, Multidimensional Stochastic Processes as Rough Paths
Gubinelli, Controlling rough paths; Hairer, Invent. Math. 2014

Next¶

Rough paths index · Stochastic processes · Martingales · SPDE · Functional analysis · Fourier analysis · Signature & Features · Controlled & Sewing · FBM & Non-Semimartingale · Branched & Hopf

Regime	Tool
\(p\in[1,2)\)	Young integration / ODE
\(p\in[2,3)\)	level-\(2\) rough paths
\(p\in[3,4)\)	level-\(3\) (need triple iterated integrals)