Lorenzo Cristofaro, Bernstein processes: an extension of fractional Brownian motion.

Abstract: The generalization of fractional Brownian motion in infinite-dimensional white noise space and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. In this talk, we present Bernstein Processes, a class of Gaussian processes defined through a class of generalized fractional operators. Our aim is to extend the fractional Brownian motion in white noise space as well as through Mandelbrot-Van Ness representation by means of more general fractional derivatives and integrals, which we define through Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein-Uhlenbeck process. We also present some properties of this class of processes (such as continuity, stationarity, persistency and anti-persistency) according to the conditions satisfied by the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.
Joint work with Prof. Beghin and Prof. Mishura.




Céline Esser, Regularity of Weighted tensorized fractional Brownian textures.

Abstract: In this presentation, we introduce a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. We study statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties. Additionally, we introduce natural functional spaces associated with these processes and propose a wavelet characterization of these spaces.





Arturo Jaramillo Gil, Some problems on matrix-valued fractional Brownian motion.

Abstract: The study of the eigenvalues of matrix-valued processes with Gaussian entries holds significant relevance across various branches of mathematics , including; operator theory free probability and statistics. A natural generalization of this type of structure consists of introducing a parameter that could allow "evolution" on the underlying matrices: one way to do this is to consider matrices with fractional Brownian motions in each entry. In this talk, we will discuss certain subtopics in the theory of matrix-valued processes: the eigenvalue collision, well-posedness of stochastic equations for the eigenvalues and functional fluctuation of linear statistics of random matrices whose entries are independent fractional Brownian motions. Moreover, a series of open problems will be presented, highlighting avenues for further research in this domain.





Yassine Nachit, Local times for systems of non-linear stochastic heat equations.

Abstract: It is widely known that in order to explore sample path properties of stochastic processes that entail rough trajectories, a common method, which was first observed by Berman, is to investigate the regularity of the trajectories of their local times. Indeed, Berman’s principle elucidates that roughness or irregularity of sample paths of a stochastic process can be reflected in the regularity (or smoothness) of the local time’s trajectories of the underlying process. In this regard, this talk continues in this vein by using local times to study sample path properties of a class of non-Gaussian processes. We consider a d-dimensional system of non-linear stochastic heat equations driven by space-time white noise. Our primary focus is on investigating the existence of the local time of the solution to this system. We establish that when the dimension, d, is less than or equal to 3, the local time exists. However, when the dimension, d, is greater than or equal to 4, the local time does not exist. Furthermore, we establish the joint continuity of the local time when it exists. These findings are instrumental in investigating the irregularity of the sample paths of the solution to this system.





Alexandre Richard, Densities of singular SDEs driven by fractional Brownian motion, and application to McKean-Vlasov equations.

Abstract: In this talk, we consider first the SDE $dX_t = b(t,X_t) + dB_t$, where $b$ is singular (e.g. a distribution) and $B$ is a fractional Brownian motion. We present and review well-posedness results about this equation, giving criteria that relate the regularity of $b$ and the parameter $H$. Then we study the time-space regularity of the density of the solution. Exploiting this regularity, we prove the existence of solutions for McKean-Vlasov equations of the form $dY_t = \mu_t \ast b(t,Y_t) + dB_t$, where $\mu_t$ is the law of the solution $Y_t$, for a drift $b$ which is allowed to be more singular than in the linear case.
Joint work with L. Anzeletti, L. Galeati and E. Tanré.




Mathieu Rosenbaum, From no arbitrage to market impact and rough volatility.

Abstract: The goal of this work is to disentangle the roles of volume and participation rate in the price response of the market to a sequence of orders. To do so, we use an approach where price dynamics are derived from the order flow via no arbitrage constraints and make connections with the rough volatility paradigm. We also introduce in the model sophisticated market participants having superior abilities to analyse market dynamics. Our results lead to two square root laws of market impact, with respect to executed volume and with respect to participation rate.
This is joint work with Bruno Durin and Grégoire Szymanski.