Brownian motion as a rough path

Question

I'm reading through the Fritz and Hairer book on Rough Path Theory and naturally want to know that a Brownian rough path or enhanced Brownian motion yields a rough path.

By the classical Kolmogorov criterium I know that for almost every $\omega\in\Omega$ the path $t\mapsto B.(\omega)$ is $\alpha$-Hölder continuous for any $\alpha<1/2$.

If I now define the second order process/object $\mathbb{B}:[0,T]^2\rightarrow \mathbb{R}$ via Itô integration, that is define

$$\mathbb{B}^{Itô}_{s,t}:=\int_s^t B_rdB_r-B_sB_{s,t}$$ where the Integral is made sense of as Itô-integral. How do I know that I obtain an $\alpha$-Rough Path?

The Hölder condition on $\mathbb{B}^{Itô}_{s,t}$ can be verified via a Kolmogorov argument and thus one only needs to verify the algebraic condition, i.e. Chen's relation.

Chen's relation for $\mathbb{B}^{Itô}_{s,t}$ now clearly holds in $L^2(\Omega)$ due to the additive nature of the stochastic integral, but I need Chen's relation to hold almost surely. How can this we shown when the integrals aren't even defined in an almost sure sense but only in $L^2(\Omega)$? Or is it possible to show the additivity of the stochastic integral to hold almost surely even though the stochastic integral is only defined in $L^2(\Omega)$?

Answer 1

This is further mentioned in "3.2 Ito Brownian motion" in the "a course on rough paths":

The $\mathbb{B}_{s,t}:=\int_{s}^{t}B_{s,r}\otimes dB_{r}$ satisfies the Chen relation on an event of probability one. They basically do this by (re)defining the increment $\mathbb{B}_{s,t}$ in terms of $\mathbb{B}_{0,\cdot}$

$$\mathbb{B}_{s,t} =\mathbb{B}_{0,t} -\mathbb{B}_{0,s} − B_s \otimes B_{s,t}.$$

This variable is defined almost surely (as is any $L^2$-Cauchy limit) and as you mentioned it satisfies the Chen relation's by breaking the integral into $\int_{s}^t=\int_{s}^u+\int_{y}^t$.

In "Lecture 27. Approximation of the Brownian rough path", the author even obtains an almost sure convergence to the above rough path in the p-variation norm by using the interpolation of Brownian motion.

As mentioned here What is the explicit obstruction to almost sure convergence in stochastic integrals? one can even have almost sure convergence in the Ito-integral.

If the meshes are chosen regularly with meshwidth $h_n\to 0$ such that $$ \sum_{n=1}^{\infty}\mathbb{E}\left[ \int_{0}^{1}(f(\omega,x)-f(\omega,\lfloor x/h_n\rfloor h_n)^{2}\;dx\right]<\infty, $$ then the Itô sums converge almost surely to the Itô integral (under usual conditions on the integrand) This is proven, for example, in Remark 7.7 of the textbook Brownian motion by Mörters and Peres. It is an immediate consequence of Itô's isometry. In particular, if $f$ is $1/2+\epsilon$ Hölder continuous, then $h_n=n^{-1}$ guarantees pathwise convergence.