Related papers: Asymptotics for a Class of Self-Exciting Point Pro…
In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection…
The problem of sums of independent, identically distributed random variables with stretched-exponential tails exhibits a dynamical phase transition and has recently reemerged in the context of active transport and condensation phenomena. We…
For one-dimensional Jump-Drift and Jump-Diffusion processes converging towards some steady state, the large deviations of a long dynamical trajectory are described from two perspectives. Firstly, the joint probability of the empirical…
This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for…
In this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations. Under the mild assumption that the so-called characteristic function of our system is completely…
Lattice models of coupled dynamical systems lead to a variety of complex behaviors. Between the individual motion of independent units and the collective behavior of members of a population evolving synchronously, there exist more…
The Hawkes self-excited point process provides an efficient representation of the bursty intermittent dynamics of many physical, biological, geological and economic systems. By expressing the probability for the next event per unit time…
Consider a one-dimensional stochastic differential equation with jumps $$\mathrm d X(t) = a(X(t))\mathrm d t + \sum_{k = 1}^m b_k(X(t-))\mathrm d Z_k(t),$$ where $Z_k, \ k \in \{1, 2, ..., m\}$ are independent centered L\'evy processes with…
We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the…
We introduce a Hawkes-like process and study its scaling limit as the system becomes increasingly endogenous. We derive functional limit theorems for intensity and fluctuations. Then, we introduce a high-frequency model for a price of a…
These short lecture notes contain a not too technical introduction to point processes on the time line. The focus lies on defining these processes using the conditional intensity function. Furthermore, likelihood inference, methods of…
The classical modeling of spatial extremes relies on asymptotic models (i.e., max-stable processes or $r$-Pareto processes) for block maxima or peaks over high thresholds, respectively. However, at finite levels, empirical evidence often…
It is shown that for a non-decreasing self-similar stochastic process $T$ with independent increments, the range of $T$ forms a Poisson point process with $\sigma$-finite intensity if and only if the one-dimensional distribution of $T(1)$…
We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that…
We study randomly stopped sums via their asymptotic scales. First, finiteness of moments is considered. To generalise this study, asymptotic scales applicable to the class of all heavy-tailed random variables are used. The stopping is…
We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns…
We consider stochastic point processes generating time series exhibiting power laws of spectrum and distribution density (Phys. Rev. E 71, 051105 (2005)) and apply them for modeling the trading activity in the financial markets and for the…
For a TASEP on $\mathbb Z$ with the step initial condition we identify limits as $t\to\infty$ of the expected total number of jumps until time $t>0$ and the expected number of active particles at a time $t$. We also connect the two…
We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are…
We investigate the two-points correlation function for several boundary-driven interacting particle systems. Our goal is to show that the time evolution of that correlation function is solution to a partial differential equation that can be…