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We derive the first-passage-time statistics of a Brownian motion driven by an exponential time-dependent drift up to a threshold. This process corresponds to the signal integration in a simple neuronal model supplemented with an…
The paper deals with disorders detection in the multivariate stochastic process. We consider the multidimensional Poisson process or the multivariate renewal process. This class of processes can be used as a description of the distributed…
We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed.…
Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad…
We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set…
We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the…
In the Wiener disorder problem, the drift of a Wiener process changes suddenly at some unknown and unobservable disorder time. The objective is to detect this change as quickly as possible after it happens. Earlier work on the Bayesian…
We present a new simple method for rounding a semidefinite programming relaxation of a constraint satisfaction problem. We apply it to the problem of approximate angular synchronization. Specifically, we are given directed distances on a…
We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $\sigma_1$ and $\sigma_2$ denote the volatilities on the negative and…
We register a stochastic sequence affected by one disorder. Monitoring of the sequence is made in the circumstances when not full information about distributions before and after the change is available. The initial problem of disorder…
We introduce a system of Brownian particles, each absorbed upon hitting an associated moving boundary. The boundaries are determined by the conditional probabilities of the particles being absorbed before some final time horizon, given the…
We study the quickest detection problem of a sudden change in the arrival rate of a Poisson process from a known value to an unknown and unobservable value at an unknown and unobservable disorder time. Our objective is to design an alarm…
We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish…
We identify an issue in recent approaches to learning-based control that reformulate systems with uncertain dynamics using a stochastic differential equation. Specifically, we discuss the approximation that replaces a model with fixed but…
We present a theory for the steady-state dynamics of a two-dimensional system of spherically symmetric active Brownian particles. The derivation of the theory consists of two steps. First, we integrate out the self-propulsions and obtain a…
For refracted skew Brownian motion (skew Brownian motion with two-valued drift), adopting a perturbation approach we find expressions of its potential densities. As applications, we recover its transition density and study its long-time…
We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general nonlinear function. The problem is inherently time inconsistent due to the Choquet integration involved. We…
A novel sequential change detection problem is proposed, in which the goal is to not only detect but also accelerate the change. Specifically, it is assumed that the sequentially collected observations are responses to treatments selected…
This paper deals with the identification of the multivariate fractional Brownian motion, a recently developed extension of the fractional Brownian motion to the multivariate case. This process is a $p$-multivariate self-similar Gaussian…
In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type f-divergence penalty treated in the general framework of a continuous…