Related papers: Nonlinear Semimartingales and Markov Processes wit…
A novel probabilistic framework for modelling anomalous diffusion is presented. The resulting process is Markovian, non-homogeneous, non-stationary, non-ergodic, and state-dependent. The fundamental law governing this process is driven by…
In this article we study a class of stochastic functional differential equations driven by L\'{e}vy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals.…
We study a general non-homogeneous Skellam-type process with jumps of arbitrary fixed size. We express this process in terms of a linear combination of Poisson processes and study several properties, including the summation of independent…
This paper shows how the theory of dynamic risk measures provides viscosity solutions to a family of second-order parabolic partial differential equations, even in the degenerate case. First, motivated by the martingale problem approach of…
By modeling the interaction of a system with an environment through a renewal approach, we demonstrate that completely positive non-Markovian dynamics may develop some unexplored non-standard statistical properties. The renewal approach is…
This paper investigates a class of multiscale stochastic control problems driven by $\alpha$-stable L\'evy noises, where the controlled dynamics evolve across separate slow and fast time scales. The associated value functions are governed…
We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions and we show that…
We show the existence of L\'evy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like with stability index functions for which the closures of the…
Given an It\^o semimartingale $X$, its Markovian projection is an It\^o semimartingale $\widehat{X}$, with Markovian differential characteristics, that matches the one-dimensional marginal laws of $X$. One may even require certain…
It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale…
We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of…
The aim of this paper is to study the fluctuations of a general class of supercritical branching Markov processes with non-local branching mechanisms. We show the existence of three regimes according to the size of the spectral gap…
We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term $-\beta$, which includes as a limit case…
Let $\mathbb{\hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $\mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $\mathcal{P}$-semimartingale under $\mathbb{\hat{E}}$. Given an open…
A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables…
Let $X$ be the unique normal martingale such that $X_0=0$ and \[\mathrm{d}[X]_t=(1-t-X_{t-}) \mathrm{d}X_t+\mathrm{d}t\] and let $Y_t:=X_t+t$ for all $t\geq 0$; the semimartingale $Y$ arises in quantum probability, where it is the…
In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general…
We study Lorentz processes in two different settings. Both cases are characterized by infinite expectation of the free-flight times, contrary to what happens in the classical Gallavotti-Spohn models. Under a suitable Boltzmann-Grad type…
We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the…
In this paper, we consider continuous-time Markov chains with a finite state space under nonlinear expectations. We define so-called Q-operators as an extension of Q-matrices or rate matrices to a nonlinear setup, where the nonlinearity is…