# fractional Brownian Motion driven stochastic integrals

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young (Link to Young's classical paper), the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

## 1 Answer

Yes, RPT allows you to define a notion of stochastic integral against fBm for a class of integrands that is larger than what Young's theory allows. Assuming that you're really interested in solving SDEs, so that $u$ locally looks again like $B^H$, you can go down to $H > {1\over 4}$. Below that things break down, and there's actually a good reason for that.

The resulting notion of integration does satisfy the usual chain rule, but moment bounds are much trickier and there's nothing as sharp as Burkholder's inequality. The best kind of moment bounds that I'm aware of are those from the article by Cass, Litterer and Lyons.

Regarding other tools, regularity structures are a generalisation of RPT, so they will give you essentially the same notion of integration, although in some special cases you can get a bit more mileage out of it because of its greater flexibility.

WNA and Skorokhod integration also give you some notion of stochastic integral that generalises Itô's integral, but this is completely different and the fact that it is a "reasonable" notion of "integral" (approximable by some kind of Riemann sums) is very specific to the case $H = {1\over 2}$.

• Dear @Martin Hairer, thank you very much for your answer. (a) I have added an edit, which clarifies my question. (b) Could you give a more detailed explanation on your last point? Specially the part with "this is completely different". – stochic Jul 6 '18 at 11:12
• If you open any book on RPT you'll get your answer to Q1. Regarding Q3 I would suggest you compute the Skorokhod integral for an elementary integrand and you'll see that it has not much in common with the corresponding Riemann sum. – Martin Hairer Jul 7 '18 at 7:13