You have seen all kinds of integration theories before — Itô, Stratonovich, and I'm sure plenty others. Rough paths takes a step back and asks what we want from an integration theory. And so long as we want three basic axioms we are forced to study the iterated integrals.

We want to create an integration theory for

$$\int_0^t F(X(r))dX(r),$$

where $X$ is a signal that is $\alpha$-Hölder continuous with $\alpha\in (1/3,1/2]$ and $F$ is a smooth function. We want our integration theory to satisfy the following three properties.

(Integral of $1$) We want $$\int_s^t 1dX(r)=X(t)-X(s).$$

(Chastles' relation) $$\int_s^u F(X(r))dX(r)+\int_u^t F(X(r)) dX(r)=\int_s^t F(X(r)) dX(r).$$

(Linearity) $$\int_s^t F_1(X(r))+cF_2(X(r))dX(r)=\int_s^t F_1(X(r))dX(r)+c\int_s^t F_2(X(r))dX(r).$$

Then for a partition of $[0,t]$, $\mathcal P=\{0=t_0<\dotsb<t_n=t\}$ we have that the integral
\begin{align}
\int_0^t F(X(r)) dX(r)&=\sum_{k=0}^n \int_{t_k}^{t_{k+1}} F(X(r))dX(r) \nonumber \\
&=\sum_{k=0}^n \int_{t_k}^{t_{k+1}} F(X(t_k))+F'(X(t_k))(X(r)-X(t_k))+O(|r-t_k|^{2\alpha})dX(r) \nonumber \\
&=\sum_{k=0}^n \biggl(F(X(t_k))(X(t_{k+1})-X(t_k))+F'(X(t_k))\int_{t_k}^{t_{k+1}}(X(r)-X(t_k)) dX(r) \nonumber \\
&~+O(|t_{k+1}-t_k|^{3\alpha})\biggr).
\end{align}
As $3\alpha>1$, the remainder term should go to $0$ as the mesh size goes to $0$. This reduces the problem of defining the integral $\int_0^t F(X(r))dX(r)$ to just defining $\int_{t_k}^{t_{k+1}}(X(r)-X(t_k)) dX(r)$.

The point is that if we want these three basic properties, the integral $\int_0^t F(X(r))dX(r)$ is defined if $\int_{t_k}^{t_{k+1}}(X(r)-X(t_k)) dX(r)$ is. Rough paths reduces defining $\int_0^t F(X(r))dX(r)$ for every $F$ to just $F$ the identity. Not only this, but it proposes a definition of integral as a standard Riemann–Stieltjes integral with a correction term.

We must construct some object $\mathbb X:\Delta_2^T\to \mathbb R^{d\times d}$ where $d$ is the dimension of the signal $X$ where $\Delta_2^T=\{(s,t)\in [0,T]^2:s\leq t\}$. The $(i,j)$th component of $\mathbb X$ represents the integral of the $i$th component of $X$ against the $j$th component.

What properties would we like the iterated integral to satisfy?

Well first it better be $2\alpha$-Hölder. That is
$$\sup_{s\neq t}\frac{|\mathbb X_{s,t}|}{|t-s|^{2\alpha}}<\infty.$$

Second, we want $\mathbb X$ to algebraically relate to $X$. What properties should it satisfy? If we want to solve differential equations, we first consider smooth $X$ and look at

$$Y_s'(t)=Y_s(t) X'(t),$$
with initial condition $Y_s(s)=1$. This has solution $$Y_s(t)=\exp(X(t)-X(s))=1+\int_s^t dX(r)+\int_s^t \int_s^{r_2} dX(r_1)dX(r_2)+\dotsb.$$

We want multiplicativity $Y_s(t)=Y_s(u)Y_u(t)$. On the level of iterated integrals we have the following relations

\begin{align*}
1&=1\\
\int_s^t dX(r)&=\int_s^u dX(r)+\int_u^t dX(r)\\
\int_s^t \int_s^{r_2} dX(r_1)dX(r_2)&=\int_s^u \int_s^{r_2} dX(r_1)dX(r_2)+\int_u^t \int_u^{r_2} dX(r_1)dX(r_2)+\int_s^u dX(r)\otimes\int_u^t dX(r)\\
&\cdots.
\end{align*}

The third equation is called Chen's relation.

A **rough path** is therefore a pair, $(X,\mathbb X):\Delta_2^T\to \mathbb R^d\oplus \mathbb R^{d\times d}$, satisfying the following properties

\begin{align*}
&X_{s,t}=X(t)-X(s)\\
&\sup_{s\neq t}\frac{|\mathbb X_{s,t}|}{|t-s|^{2\alpha}}<\infty\\
&\mathbb X_{s,t}=\mathbb X_{s,u}+\mathbb X_{u,t}+X_{s,u}\otimes X_{u,t}.
\end{align*}

The second order process can always be defined (this is the Lyons-Victoir extension theorem), although the construction given by Lyons-Victoir is abstract.

However, the second order process $\mathbb X$ is not unique as given any rough path $(X_{s,t},\mathbb X_{s,t})$ and any $f\in C^{2\alpha}$ we have that $(X_{s,t},\mathbb X_{s,t}+f(t)-f(s))$ is also a rough path (just check the properties). Conversely, any two rough paths differ by the addition of the increment of some $f\in C^{2\alpha}$. Therefore rough paths give a way of "parameterizing" integration theories that satisfy those three axioms through this function $f$.