3.15.4 Rough Path Theory

Basic Prerequisites: 02 · The Signature Algebra and Brownian Motion & Martingales. The Itô-integral side is Itô Integral & Itô’s Lemma.

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1. Intuition & Practical Objective

Everything in pages 01–03 assumed the path is nice enough for its iterated integrals to be classically defined. Finance’s paths are not: realised volatility, spot, order flow and high-frequency price are all at least as rough as Brownian motion - and rough vol is rougher still. For such paths the naive iterated integrals do not converge: they blow up, and the whole signature construction looks like it should fail. Rough path theory (Lyons 1998; Friz–Victoir 2010) is the precise answer to “what actually happens, and how do we make it work?” Its three pillars:

1. The $p$-variation norm. Roughness is not a binary fact but a number: the smallest $p\ge 1$ such that the $p$-variation ${\sum }_k|\Delta X_{t_k}{|}^p$ stays finite. Smooth paths have $p=1$; Brownian motion has $p=2$ (its quadratic variation is finite, its linear variation is not); fractional noise can have $p>2$. The $p$-variation is the correct topology for controlling iterated integrals.
2. The rough-path lift. When the path is too rough for level-2 integrals to be classically well defined, the resolution is to declare the level-2 object (the Lévy area) as part of the data. A geometric rough path is the package $\mathbf{X}=(1,X,{\mathbb{X}}^2,\dots )$ of the path and its consistently-chosen iterated integrals, satisfying Chen’s identity and the right analytic bounds. This is the “lift” - a canonical, stable choice (e.g. the Stratonovich area for Brownian motion).
3. The extension theorem. Given the lift to level $\lfloor p\rfloor$ (e.g. level 2 for a $p$-variation path with $p<3$), the extension theorem (Lyons) says the whole signature is determined and can be constructed recursively - the higher iterated integrals are functions of the lower levels, not new free choices. This is what makes the signature computable for rough paths: choose the area, and Chen’s identity + the bounds extend it to all levels.

The practical objective: know that signatures are not just “for nice paths,” know the role of $p$-variation in making them rigorous, and understand the lift as the single free, canonical choice (the area) that everything else is built from.

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2. Mathematical Ground Truth & Derivations

2.1 $p$-variation and what it measures

The $p$-variation of $X:[0,T]\to {\mathbb{R}}^d$ is

$$\Vert X{\Vert }_{p\text{-var}}=(\underset{\text{partitions}}{\sup }\sum _k|X_{t_{k+1}}-X_{t_k}{|}^p{)}^{1/p},$$

finite for smooth paths at $p=1$ (total variation), finite for Brownian motion at $p=2$ (and only there, $p>2$), finite for fractional Brownian motion with Hurst $H$ at any $p>1/H$. The defining empirical fact, verified in §3:

• Brownian: the quadratic variation $\sum |\Delta B{|}^2\to T$ (finite, nonzero), the linear variation diverges, the cubic variation $\to 0$.
• Smooth: linear variation is finite, and every higher $p$-variation $\to 0$.

So Brownian motion is exactly the boundary $p=2$ - the reason the theory is organised around $p$-variation and why level-2 (the area) is the first “extra” piece the lift must supply.

2.2 Why level-2 is the hard one: the rough-path lift

For a path of finite $p$-variation with $2\le p<3$, the level-1 signature is classically defined but the level-2 iterated integral is not (it depends on the refining sequence - Itô vs Stratonovich differ). The rough-path solution is to choose the area ${\mathbb{X}}_{s,t}^{ij}$ satisfying the analytic bound

$$|{\mathbb{X}}_{s,t}^{ij}|\le C|t-s{|}^{2/p}$$

and Chen’s identity, and treat $\mathbf{X}=(X,\mathbb{X})$ as the fundamental object. For Brownian motion the canonical choice is the Stratonovich area; §3 shows it is a finite, converging quantity under refinement - a well-defined rough-path lift - exactly where the raw level-1 variation blows up. The choice of area is the one free, non-redundant degree of freedom of the lift; everything above is determined.

2.3 The extension theorem

The central result of Lyons 1998: a multiplicative functional $\mathbf{X}=(1,X,{\mathbb{X}}^2,\dots ,{\mathbb{X}}^{\lfloor p\rfloor })$ satisfying Chen’s identity and the analytic bounds extends uniquely to a full signature $\mathbf{X}=(1,X,{\mathbb{X}}^2,{\mathbb{X}}^3,\dots )$. Each higher level is constructed by an explicit limiting procedure from the lower ones, and the extension is continuous in the $p$-variation topology. Two consequences that matter:

• Uniqueness of the extension: the higher iterated integrals are not free; given the lift to level $\lfloor p\rfloor$ there is exactly one geometric signature.
• Continuity / computability: the signature of a rough path is the limit of the signatures of its (smooth) approximations - which is precisely why the explicit piecewise-linear sums in pages 01–03 are legitimate: for a geometric rough path they converge to the rough-path signature (Friz–Victoir). This is the bridge between the “nice path” algebra and the rough reality.

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3. Computational Implementation - p-variation of Brownian vs smooth, and a stable rough-path lift

We (i) compute the partition $p$-variation of a seeded Brownian path at successively finer grids for $p=1,2,3$ and show the three regimes (diverge / converge to $T$ / vanish), (ii) do the same for a smooth sine path (all higher $p$-variation $\to 0$), and (iii) compute the Lévy area of a 2D Brownian path under refinement, showing it is bounded and converges to a finite limit - a well-defined level-2 rough-path lift - where the $p<2$ variation diverges. Stdlib only, seeded (random.seed), deterministic.

Reading the output.

• Three regimes, one path. For the same Brownian sample: the $p=1$ variation diverges ($35.9\to 100.9$ as the grid refines), the $p=2$ variation converges to $T=1$ ($1.014\to 0.994$, finite and nonzero - the classic quadratic variation), and the $p=3$ variation collapses toward $0$ ($0.0365\to 0.0124$). This is the exact “Brownian has finite variation iff $p\ge 2$” statement made visible.
• Smooth paths are the opposite. The sine path has finite $p=1$ variation ($4.00$, the total variation of $\sin 2\pi t$) that does not diverge, and its higher $p$-variations vanish with refinement ($p=2$: $1.97\times 10^{-1}\to 1.23\times 10^{-2}$; $p=3$: $1.05\times 10^{-2}\to 4.11\times 10^{-5}$). Smoothness and Brownian roughness are separated by exactly the $p$-variation index.
• The rough-path lift is stable where the variation is not. The 2D Brownian Lévy area stays bounded and converges under refinement ($-0.106\to -0.093$, increments shrinking), i.e. it is a well-defined level-2 rough-path lift - even though the same path’s $p<2$ variation blows up. This is the concrete content of the lift: you choose the area, and it is a finite, canonical object, whereas the naive level-2 integral over a rough path would depend on the discretisation convention (Itô vs Stratonovich).

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4. Failure Modes & First-Principles Breakdowns

1. Assuming iterated integrals of financial paths “just converge.” They do not: at Brownian roughness the level-2 integral is not classically defined and depends on the refinement convention. Every signature computation on real data implicitly selects a rough-path lift; failing to make it explicit (Itô vs Stratonovich) is a silent source of arbitrariness. The geometric (Stratonovich / piecewise-linear) convention used throughout this folder is the canonical choice (Friz–Victoir).
2. Treating $p$-variation as a fixed constant of a model. It is a norm, and different co-ordinates of a system can have different roughness (e.g. spot vs its realised vol). The extension theorem needs the correct $p$ for the whole lift; underestimating $p$ overstates the control and invalidates the bounds.
3. Believing the lift is “just an integral.” The level-2 area is a choice (with the Itô/Stratonovich ambiguity) before it is a computation. For non-geometric data (raw Itô-increment series) the chosen area may not satisfy the shuffle identity - the reason augmentation/lead-lag is required before signatures are meaningful (§05).
4. Overlooking that uniqueness holds up to tree-like equivalence for rough paths. The extension theorem gives a unique extension given the lower lift, but the signature as a whole determines the rough path only up to tree-like reparametrisation (Boedihardjo et al. 2016) - the same caveat as §03, sharpened for the rough case.
5. Ignoring the discretisation-vs-lift distinction in Monte Carlo. In simulation (§03 of Numerical Methods · 03 Monte Carlo) a fine Euler grid does not automatically converge to the geometric rough path; if the payoff depends on the area (as variance/vol products do), the scheme must target the lifted object, not the naked increments.

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5. References

• Lyons, Terry J. (1998), Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14(2), 215–310
• Lyons, T., Caruana, M., Lévy, T. (2007), Differential Equations Driven by Rough Paths, Springer LNM 1908
• Friz, P. K. & Victoir, N. (2010), Multidimensional Stochastic Processes as Rough Paths, Cambridge Studies in Adv. Math. 120
• Friz, P. K. & Hairer, M. (2014), A Course on Rough Paths (Springer)
• Boedihardjo, H., Geng, X., Lyons, T., Yang, D. (2016), The signature of a rough path: uniqueness, Advances in Mathematics 293
• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough

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6. Connected Graph Bridges

• Back: 03 · Log-Signature & Uniqueness
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions · Index Hub
• Theory: Brownian Motion & Martingales · Itô Integral & Itô’s Lemma · Stochastic Calculus & Itô
• Applied: Rough Volatility · Numerical Methods · 03 Monte Carlo