Rough paths theory - why is it natural?

Question

First a disclaimer, I know very little rigorously about the subject and I may have many misconceptions in what follows so please correct me if I’m wrong about anything. Also none of this is meant to be a critique of the theory. I’m not qualified to do so in the first place, and I’m very sure I’m missing something fundamental here.

But anyway, the main thing I want to ask is “how is rough paths not cheating?”. From what I understand, some paths are too rough to be integrated in the usual sense, in their words they don’t carry enough “data” by themselves to do calculus. So you just... postulate additional data, namely their behaviour under iterated integrals and then you can do it. Morally, why is this okay?

In what way is this theory then a natural extension of probability theory instead of just giving ourselves additional structure that didn’t exist in the first place?

Answer 1

The phrase "postulating" might be a bit misleading. Often, the iterated integrals (the second order process) are defined in some canonical way. Besides, all rough path lifts differ by the increment of a function.

Lemma: Let $(X,\mathbb X)$ and $(X,\tilde{\mathbb X})$ be two $\gamma$-rough paths. Then there is some $F\in C^{2\gamma}$ so that $\mathbb X_{s,t}=\tilde{\mathbb X}_{s,t}+F(t)-F(s).$

Proof: Chen's relation says that $\mathbb X_{s,t}-\mathbb X_{s,u}-\mathbb X_{u,t}=\tilde{\mathbb X}_{s,t}-\tilde{\mathbb X}_{s,u}-\tilde{\mathbb X}_{u,t}$ for all $s,u, t.$ Defining $F_{s,t}=\mathbb X_{s,t}-\tilde{\mathbb X}_{s,t}$ shows that $F_{s,t}-F_{s,u}-F_{u,t}=0$ or $F_{s,t}=F_{s,u}+F_{u,t}$. Chen's relation also implies that $F_{s,u}=-F_{u,t}$. So $F_{s,t}=F_{u,t}-F_{u,s}$. Also, $F\in C^{2\gamma}$ because $\mathbb X,\tilde{\mathbb X}$ are. Define $F(t):=F_{u,t}$.

Note crucially that in general $F$ cannot be a smooth function of $X$. If $F(t)=\phi(X(t))$ for smooth $\phi$ then $F(t)-F(s)=\phi'(X(s^\ast))(X(t)-X(s))$ by the mean value theorem. So you should expect no more regularity from $F$ than from $X$. For example Ito and Stratonovich differ by $F(t)=\frac12 t$.

This is reminiscent of standard calculus. If you have a continuous function $f$ then you can ``postulate" any antiderivative $F$ you want, so long as it satisfies $F'=f$. There are infinitely many choices. However given any $F_1,F_2$ that satisfy $F_1'=F_2'=f$, we have that $F_1-F_2=C$ for some constant $C$.

One way of stating this is in terms of a ``difference" operator $\delta$. Let $V$ be a vector space. Given a function $F:[0,1]\to V$ we can define $\delta F:[0,1]^2\to V$ by $\delta F_{s,t}=F(t)-F(s)$. Furthermore, if $F:[0,1]^2\to V$ we define $\delta F:[0,1]^3\to V$ by $\delta F_{s,u,t}=F_{s,t}-F_{s,u}-F_{u,t}$.

With this difference operator we have that $\delta F_{s,t}=0$ iff $F$ is constant and $\delta F_{s,u,t}=0$ iff $F=\delta G$ for some $G:[0,1]\to V$.

The point I am making is that you can't write down ``whatever you want" - in practice it is quite hard to explicitly construct natural rough paths above various signals.

What happens in practice is that you take a natural smooth approximation of $X$, call it $X^\varepsilon$. You construct the Riemann-Stieltjes integrals above $X^\varepsilon$ and you hope that this pair converges in the rough topology. Sometimes it does sometimes it doesn't. When it does converge, such limiting rough paths are called ``geometric."