Let $L = (L(t))_{t \geq 0}$ be a multivariate Lévy process with Lévy measure $\nu(dy) = \exp(-f(|y|))dy$ for a smoothly regularly varying function $f$ of index $\alpha > 1$. The process $L$ is renormalized as $X^\varepsilon(t) = \varepsilon L(\tau_\varepsilon t),\ t \in [0,T]$, for a scaling parameter $\tau_\varepsilon = o(\varepsilon^{-1})$, as $\varepsilon \to 0$. We study the behavior of the bridge $Y^{\varepsilon,x}$ of the renormalized process $X^\varepsilon$ conditioned on the event $X^\varepsilon(T) = x$ for a given end point $x \neq 0$ and end time $T > 0$ in the regime of small $\varepsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\varepsilon,x}$ with a specific speed function $S(\varepsilon)$ and an entropy-type rate function $I_x$ on the Skorokhod space in the limit $\varepsilon \to 0+$. We show that the asymptotic energy minimizing path of $Y^{\varepsilon,x}$ is the linear parametrization of the straight line between 0 and $x$, while all paths leaving this set are exponentially negligible. We also infer a LDP for the asymptotic number of jumps and establish asymptotic normality of the jump increments of $Y^{\varepsilon,x}$. Since on these short time scales ($\tau_\varepsilon = o(\varepsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^\varepsilon(t),\ t \in [0,T]$, for which we solve a specific nonlinear functional equation.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin. Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin. Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.

In a first step, I will present a probabilistic interpretation for a nonlinear, nonlocal PDE on $(0,\infty)$ introduced to model cell polarization. More precisely, the solution of the PDE is the density at time t of a certain diffusion process. This PDE includes a nonlinear boundary attraction term, which translates on the process into a drift term depending on the density of the solution at $x=0$. Then, this nonlinear process can be approximated by a particle system by replacing the value of the density at 0 by an empirical mean of the local times at the edges of N copies of the process. This particle system can be seen as an N-dimensional process living in a non-regular open and with an oblique reflection on the boundary. I will discuss the existence for the particle system, for the nonlinear process and for the PDE and the links between these three objects. In particular, for a sufficiently large attraction coefficient, these objects must necessarily blow-up in finite time.

In a second step I will study the long term behavior of a class of Self-Interacting Diffusions on $\mathbb R$, where the particle is attracted or repelled by its past trajectory, with aging and linear interactions with past positions. The novelty of this model is to introduce a weight depending on past trajectories through a memory kernel to describe the aging phenomenon. In a first step, we study the case where the memory kernel is general of convolution type. We observe three distinct asymptotic regimes when time becomes large depending on the data and the first moment of the integral kernel. We then study a more realistic non-convolution model and prove a similar transition between the first two regimes. At last, we study the particular case of an exponential memory kernel which allows to construct an explicit solution.

In this work, we deal with the symmetric exclusion process with $k$ slow bonds equally spaced in the torus with $kn$ sites, where the strength of a slow bond is $α n^{-β}$, where $β>1$. For $k$ fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP’13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^θ$, where $2< θ<1+β$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If $k$ is fixed and $θ=1+β$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $θ=1+β$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).