Fractional Stochastic Calculus via Stochastic Sewing

The thesis explores stochastic calculus for fractional Brownian motion. Our approach builds upon a novel technique called stochastic sewing, originally introduced by Lê [Electron. J. Probab. 25:1-55, 2020]. The stochastic sewing has been effectively used to obtain sharp estimates on stochastic Riemann sums. The main result of the thesis is an extension of Lê’s stochastic sewing, which we refer to as the shifted stochastic sewing. This extension takes advantage of asymptotic decorrelation in stochastic Riemann sums and can be seen as a combination of Lê’s stochastic sewing and the asymptotic independence formulated by Picard [Ann. Probab. 36(6): 2235-2279, 2008]. As applications of the shifted stochastic sewing, we address two important problems in fractional stochastic calculus. Firstly, we characterize the local time of the fractional Brownian motion via level crossings, extending the classical work of Lévy to the fractional setting. Secondly, we establish the pathwise uniqueness of Young and rough differential equations driven by fractional Brownian motion. This result optimizes the regularity of the noise coefficient, which is consistent with the Brownian setting. Additionally, we demonstrate strong regularization by fractional noise for differential equations with integrable drifts. This result can be viewed as a fractional analogue of the celebrated work by Krylov and Röckner [Probab. Theory Relat. Fields 131: 154–196, 2005]

Persistence exponent for random processes in Brownian scenery

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian and non-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup\_{t\in[0,T]} \Delta\_t \leq 1]$ where $\Delta\_t = \int\_{\mathbb{R}} L\_t(x) \, dW(x).$ Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L\_t(x); x\in\mathbb{R},t\geq 0}$ is the local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent

A fractional Brownian field indexed by L2L^2 and a varying Hurst parameter

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as L\'evy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and H\"older regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on L\'evy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local H\"older regularity on general indexing collections

Small deviations for fractional stable processes

Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter H > 0. Consider ||.|| a translation invariant, b-self-similar, and p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c = (H - b - 1/p), then x log P [ log ||R|| - k 0, with k finite in the Gaussian case a = 2. If a < 2, we prove that k is finite when R is continuous and H > (b + 1/p + 1/a). We also show that under the above assumptions, x log P [ log ||X|| - k 0, where k is finite and X is the linear a-stable fractional motion with Hurst parameter 0 < H < 1 (if a = 2, then X is the classical fractional Brownian motion). These general results recover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and Non-Gaussian frameworks.Comment: 30 page

A functional approach for random walks in random sceneries

A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.Comment: 23