Asymptotical properties of distributions of isotropic Lévy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1.The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi \left(\lambda \right)?\frac{\lambda }{2}{\psi }^{\prime }\left(\lambda \right)$ varies regularly, as $\frac{t\psi {\left(r^{?1}\right)}^2}{\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)}\to 0$ the tail probability $–(\left|X_t\right|\ge r)$ is asymptotically equal to a constant times $t(\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)).$     

A one-dimensional version of the random interlacements

We base ourselves on the construction of the two-dimensional random interlacements (Comets et al., 2016) to define the one-dimensional version of the process. For this, we consider simple random walks conditioned on never hitting the origin. We compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements’ local time and the convergence in law of the local times of the conditional walk on the ring graph to the interlacements’ local times.     

Representations of max-stable processes via exponential tilting

The recent contribution (Dieker and Mikosch, 2015) obtained representations of max-stable stationary Brown–Resnick process ${\zeta }_Z\left(t\right),t\in {\mathbb{R}}^d$ with spectral process $Z$ being Gaussian. With motivations from Dieker and Mikosch (2015) we derive for general $Z$, representations for ${\zeta }_Z$ via exponential tilting of $Z$. Our findings concern Dieker–Mikosch representations of max-stable processes, two-sided extensions of stationary max-stable processes, inf-argmax representation of max-stable distributions, and new formulas for generalised Pickands constants. Our applications include conditions for the stationarity of ${\zeta }_Z$, a characterisation of Gaussian distributions and an alternative proof of Kabluchko’s characterisation of Gaussian processes with stationary increments.     

Asymptotic results for exponential functionals of Lévy processes

The asymptotic behavior of expectations of some exponential functionals of a Lévy process is studied. The key point is the observation that the asymptotics only depend on the sample paths with slowly decreasing local infimum. We give not only the convergence rate but also the expression of the limiting coefficient. The latter is given in terms of some transformations of the Lévy process based on its renewal function. As an application, we give an exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.     

Semimartingale: Itô or not ?

Itô semimartingales are the semimartingales whose characteristics are absolutely continuous with respect to Lebesgue measure. We study the importance of this assumption for statistical inference on a discretely sampled semimartingale in terms of the identifiability of its characteristics, their estimation, and propose tests of the Itô property against the non-Itô alternative when the observed semimartingale is continuous, or discontinuous with finite activity jumps, and under a number of technical assumptions.     

Stability problems for Cantor stochastic differential equations

We consider driftless stochastic differential equations and the diffusions starting from the positive half line. It is shown that the Feller test for explosions gives a necessary and sufficient condition to hold pathwise uniqueness for diffusion coefficients that are positive and monotonically increasing or decreasing on the positive half line and the value at the origin is zero. Then, stability problems are studied from the aspect of Hölder-continuity and a generalized Nakao–Le Gall condition. Comparing the convergence rate of Hölder-continuous case, the sharpness and stability of the Nakao–Le Gall condition on Cantor stochastic differential equations are confirmed. Furthermore, using the Malliavin calculus, we construct a smooth solution to degenerate second order Fokker–Planck equations under weak conditions on the coefficients.     

Distribution dependent SDEs for Landau type equations

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup $P_t^1$ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup $P_t^1$ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.     

Stochastic Komatu–Loewner evolutions and BMD domain constant

For Komatu–Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of Schramm (2000) to find the SDEs satisfied by the induced motion $\xi \left(t\right)$ on $?\mathbb{H}$ and the slit motion $\mathbf{s}\left(t\right)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogeneous functions.Next with solutions of such SDEs, we study the corresponding stochastic Komatu–Loewner evolution, denoted as ${\mathrm{SKLE}}_{\alpha ,b}$. We introduce a function $b_{\mathrm{BMD}}$ measuring the discrepancy of a standard slit domain from $\mathbb{H}$ relative to BMD. We show that ${\mathrm{SKLE}}_{\sqrt{6},?b_{\mathrm{BMD}}}$ enjoys a locality property.     

Conjugate processes: Theory and application to risk forecasting

Many dynamical phenomena display a cyclic behavior, in the sense that time can be partitioned into units within which distributional aspects of a process are homogeneous. In this paper, we introduce a class of models – called conjugate processes – allowing the sequence of marginal distributions of a cyclic, continuous-time process to evolve stochastically in time. The connection between the two processes is given by a fundamental compatibility equation. Key results include Laws of Large Numbers in the presented framework. We provide a constructive example which illustrates the theory, and give a statistical implementation to risk forecasting in financial data.     

Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-${?}^2$ summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ${?}^2$ summable. Finally, if we weaken the condition for DU, asking for coincidence on the future $\sigma$-algebra for almost every pair of pasts, we obtain a condition that is equivalent to $\beta$-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-${?}^2$ summability condition on the kernel is equivalent to the $\beta$-mixing of the compatible stationary chain.