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We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves…
Fractional-order SIR models have become increasingly popular in the literature in recent years, however unlike the standard SIR model, they often lack a derivation from an underlying stochastic process. Here we derive a fractional-order…
Dirichlet processes and their extensions have reached a great popularity in Bayesian nonparametric statistics. They have also been introduced for spatial and spatio-temporal data, as a tool to analyze and predict surfaces. A popular…
Selecting input variables or design points for statistical models has been of great interest in adaptive design and active learning. Motivated by two scientific examples, this paper presents a strategy of selecting the design points for a…
We show that bootstrap methods based on the positivity of probability measures provide a systematic framework for studying both synchronous and asynchronous nonequilibrium stochastic processes on infinite lattices. First, we formulate…
We propose a novel approach to estimate the Cox model with temporal covariates. Our new approach treats the temporal covariates as arising from a longitudinal process which is modeled jointly with the event time. Different from the…
Many stochastic physical systems evolve smoothly over time in the sense that the distribution of states changes regularly across time steps. The transition from current state to the next state can often be modeled as the combination of a…
Motivated by certain problems of statistical physics we consider a stationary stochastic process in which deterministic evolution is interrupted at random times by upward jumps of a fixed size. If the evolution consists of linear decay, the…
A continuous time mixed state branching process is constructed as the scaling limits of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic…
We introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of…
We consider the drift and diffusion properties of periodically driven renewal processes. These processes are defined by a periodically time dependent waiting time distribution, which governs the interval between subsequent events. We show…
The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of stochastic time series with time-dependent variance like it appears on a wide broad of systems besides economics in which ARCH was born. Although the ARCH process…
We present a general approach for studying autoregressive categorical time series models with dependence of infinite order and defined conditional on an exogenous covariate process. To this end, we adapt a coupling approach, developed in…
The paper is devoted to the study of the short rate equation of the form $$ d R(t)=F(R(t)) dt+\sum_{i=1}^{d}G_i(R(t-)) dZ_i(t), \quad R(0)=x\geq 0,\quad t>0, $$ with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy process…
In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we…
Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For $(X_t)$ a two-sided infinite $\alpha$-stable moving average (MA),…
We propose a general framework for studying jump-diffusion systems driven by both Gaussian noise and a jump process with state-dependent intensity. Of particular natural interest are the jump locations: the system evaluated at the jump…
In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given…
A family of continuous-time generalized autoregressive conditionally heteroscedastic processes, generalizing the $\operatorname {COGARCH}(1,1)$ process of Kl\"{u}ppelberg, Lindner and Maller [J. Appl. Probab. 41 (2004) 601--622], is…
We investigate precursors and predictability of extreme increments in a time series. The events we are focusing on consist in large increments within successive time steps. We are especially interested in understanding how the quality of…