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Dissemination
Bibliography

## Section: New Results

### Stochastic integration with respect to the Rosenblatt process.

Participant : Benjamin Arras.

From a theoretical perspective to more concrete applications, fractional Brownian motion (fbm) is a fruitful and rich mathematical object. From its stochastic analysis, initiated during the nineties, several theories of stochastic integration have emerged so far. Indeed, fbm is, in general, not a semimartingale neither a Markov process. These theories rely on different properties of the stochastic integrator process and are then of different natures. Despite the quite large number of these strategies, we can group them into two fundamentally distinct categories: the pathwise and the probabilistic approaches. The probabilistic one requires highly evolved stochastic analysis tools. Indeed, the Malliavin calculus as well as Hida's distribution theory have been used in order to define stochastic integration with respect to fractional Brownian motion ( [56] , [52] ) and more general Gaussian processes ([47] ). Moreover, fbm belongs to an important class of stochastic processes, namely, the Hermite processes. This class appears in non-central limit theorems for processes defined as integrals or partial sums of non-linear functionals of stationary Gaussian sequences with long-range dependence (see [57] ). They admit the following representation for all $d\ge 1$:

$\begin{array}{c}\hfill \forall t>0\phantom{\rule{1.em}{0ex}}{Y}_{t}^{H,d}=c\left({H}_{0}\right){\int }_{ℝ}...{\int }_{ℝ}\left({\int }_{0}^{t}\prod _{j=1}^{d}{\left(s-{x}_{j}\right)}_{+}^{{H}_{0}-1}ds\right)d{B}_{{x}_{1}}...d{B}_{{x}_{d}}\end{array}$

where $c\left({H}_{0}\right)$ is a normalizing constant such that $𝔼\left[|{Y}_{1}^{H,d}{|}^{2}\right]=1$ and ${H}_{0}=\frac{1}{2}+\frac{H-1}{d}$ with $H\in \left(\frac{1}{2},1\right)$. For $d=1$, one recovers fractional Brownian motion. These processes share many properties with fbm. Namely, they are $H$-self-similar processes with stationary increments. They possess the same covariance structure, exhibit long range-dependence and their sample paths are almost-surely $\delta$-Hölder continuous, for every $\delta . For $d=2$, the process is called the Rosenblatt process. This process has received lots of interest in the past and more recent years. Stochastic calculus with respect to the Rosenblatt process has been developped in [73] from both, the pathwise type calculus and Malliavin calculus points of view. Even if these two approaches are successful in order to define a stochastic integral with respect to the Rosenblatt process, the Malliavin calculus one fails to give an Itô's formula for the Rosenblatt process in the divergence sense. In [42] , by means of white noise distribution theory, we obtain the following result:

Theorem: Let $\left(a,b\right)\in {ℝ}_{+}^{*}$ such that $a\le b<\infty$. Let $F$ be an entire analytic function of the complex variable verifying:

$\exists N\in ℕ,\exists C>0,\forall z\in ℂ\phantom{\rule{1.em}{0ex}}|F\left(z\right)|\le C{\left(1+|z|\right)}^{N}exp\left(\frac{1}{\sqrt{2}{b}^{H}}|\Im \left(z\right)|\right)$

Then, we have in ${\left(S\right)}^{*}$:

$F\left({X}_{b}^{H}\right)-F\left({X}_{a}^{H}\right)={\int }_{a}^{b}{F}^{\left(1\right)}\left({X}_{t}^{H}\right)\diamond {\stackrel{˙}{X}}_{t}^{H}dt+\sum _{k=2}^{\infty }\left(H{\kappa }_{k}\left({X}_{1}^{H}\right){\int }_{a}^{b}\frac{{t}^{Hk-1}}{\left(k-1\right)!}{F}^{\left(k\right)}\left({X}_{t}^{H}\right)dt+{2}^{k}{\int }_{a}^{b}{F}^{\left(k\right)}\left({X}_{t}^{H}\right)\diamond {\stackrel{˙}{X}}_{t}^{H,k}dt\right),$

where $\left\{{X}_{t}^{H}\right\}=\left\{{Y}_{t}^{H,2}\right\}$, $\left\{{\stackrel{˙}{X}}_{t}^{H}\right\}$ is the Rosenblatt noise, $\left\{{\kappa }_{k}\left({X}_{1}^{H}\right);k\ge 2\right\}$ the non-zero cumulants of the Rosenblatt distribution, $\diamond$ the Wick product and $\left\{\left\{{X}_{t}^{H,k}\right\}:k\ge 2\right\}$ a sequence of processes defined by:

$\begin{array}{c}\hfill \forall t\ge 0\phantom{\rule{1.em}{0ex}}{X}_{t}^{H,k}={\int }_{ℝ}{\int }_{ℝ}\underset{k-1×{\otimes }_{1}}{\underbrace{\left(...\left(\left({f}_{t}^{H}{\otimes }_{1}{f}_{t}^{H}\right){\otimes }_{1}{f}_{t}^{H}\right)...{\otimes }_{1}{f}_{t}^{H}\right)}}\left({x}_{1},{x}_{2}\right)d{B}_{{x}_{1}}d{B}_{{x}_{2}}\end{array}$

with ${f}_{t}^{H}\left({x}_{1},{x}_{2}\right)=c\left(H\right){\int }_{0}^{t}{\prod }_{j=1}^{2}{\left(s-{x}_{j}\right)}_{+}^{\frac{H}{2}-1}ds$ and ${\otimes }_{1}$ is the contraction of order 1.

Moreover, in the same setting, we obtain the following "isometry" result for the Rosenblatt noise integral of sufficiently "good" integrand processes:

Theorem: Let $\left\{{\phi }_{t};t\in I\right\}$ be a stochastic process such that for all $t\in I$ ($I$ an interval), ${\phi }_{t}\in \left({L}^{2}\right)$ and such that the Rosenblatt noise integral of $\left\{{\phi }_{t}\right\}$ exists in ${\left(S\right)}^{*}$. Moreover, let us assume that:

$\begin{array}{c}\hfill \sum _{m=0}^{+\infty }\left(m+2\right)!{\int }_{I}{\int }_{I}{|t-s|}^{2\left(H-1\right)}<{f}_{m}\left(.,t\right);{f}_{m}\left(.,s\right){>}_{{L}^{2}\left({ℝ}^{m}\right)}dtds<+\infty ,\end{array}$

where ${\phi }_{t}={\sum }_{m=0}^{+\infty }{I}_{m}\left({f}_{m}\left(.,t\right)\right)$. Thus, we have:

$\begin{array}{cc}& 𝔼\left[{\left({\int }_{I}{\phi }_{t}\diamond {\stackrel{˙}{X}}_{t}^{H}dt\right)}^{2}\right]=H\left(2H-1\right){\int }_{I}{\int }_{I}{|t-s|}^{2\left(H-1\right)}𝔼\left[{\phi }_{t}{\phi }_{s}\right]dsdt\hfill \\ & +4\sqrt{\frac{H\left(2H-1\right)}{2}}{\int }_{I}{\int }_{I}{|t-s|}^{H-1}𝔼\left[{D}_{\sqrt{d\left(H\right)}{\delta }_{s}\circ {I}_{+}^{\frac{H}{2}}}\left({\phi }_{t}\right){D}_{\sqrt{d\left(H\right)}{\delta }_{t}\circ {I}_{+}^{\frac{H}{2}}}\left({\phi }_{s}\right)\right]dsdt\hfill \\ & +{\int }_{I}{\int }_{I}𝔼\left[{\left({D}_{\sqrt{d\left(H\right)}{\delta }_{s}\circ {I}_{+}^{\frac{H}{2}}}\right)}^{2}\left({\phi }_{t}\right){\left({D}_{\sqrt{d\left(H\right)}{\delta }_{t}\circ {I}_{+}^{\frac{H}{2}}}\right)}^{2}\left({\phi }_{s}\right)\right]dsdt,\hfill \end{array}$

where ${D}_{\sqrt{d\left(H\right)}{\delta }_{s}\circ {I}_{+}^{\frac{H}{2}}}$ is the derivative operator in the direction $\sqrt{d\left(H\right)}{\delta }_{s}\circ {I}_{+}^{\frac{H}{2}}$.

Finally, in the last section of [42] , we compare our approach to the one of [73] . More specifically, we prove that the stochastic integral with respect to the Rosenblatt process built using Malliavin calculus corresponds with the Rosenblatt noise integral when both of them exist.

Proposition: Let $\left\{{\phi }_{t};t\in \left[0;T\right]\right\}$ be a stochastic process such that $\phi \in {L}^{2}\left(\Omega ;ℋ\right)\cap {L}^{2}\left(\left[0,T\right];{𝔻}^{2,2}\right)$ and $𝔼\left[{\int }_{0}^{T}{\int }_{0}^{T}||{D}_{{s}_{1},{s}_{2}}{\phi ||}_{ℋ}^{2}\right]d{s}_{1}d{s}_{2}<\infty$ where

$\begin{array}{c}\hfill ℋ=\left\{f:\left[0;T\right]\to ℝ;{\int }_{0}^{T}{\int }_{0}^{T}f\left(s\right)f\left(t\right)|t-s{|}^{2H-2}dsdt<\infty \right\}.\end{array}$

Then, $\left\{{\phi }_{t}\right\}$ is Skorohod integrable and ${\left(S\right)}^{*}$-integrable with respect to the Rosenblatt process, ${\left\{{Z}_{t}^{H}\right\}}_{t\in \left[0;T\right]}$, and we have:

$\begin{array}{c}\hfill {\int }_{0}^{T}{\phi }_{t}\delta {Z}_{t}^{H}={\int }_{0}^{T}{\phi }_{t}\diamond {\stackrel{˙}{Z}}_{t}^{H}dt\end{array}$