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Robert Dalang

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Publications associées (33)

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A Quickest Detection Problem With An Observation Cost

Robert Dalang

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. Th ...

Inst Mathematical Statistics2015

Moments And Growth Indices For The Nonlinear Stochastic Heat Equation With Rough Initial Conditions

Robert Dalang, Le Chen

We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but th ...

Inst Mathematical Statistics2015

Stochastic integrals for spde's: A comparison

Robert Dalang

We present the Walsh theory of stochastic integrals with respect to martingale measures, and various extensions of this theory, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and ...

Elsevier2011

Predicting The Ultimate Supremum Of A Stable Levy Process With No Negative Jumps

Robert Dalang

Given a stable Levy process X = (X-t)(0

2011

On the density of the supremum of the solution to the linear stochastic heat equation

Robert Dalang, Fei Pu

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoot ...

SPRINGER2020

Random field solutions to linear SPDEs driven by symmetric pure jump Levy space-time white noises

Robert Dalang, Thomas Marie Jean-Baptiste Humeau

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Levy white noise, with symmetric alpha-stable Levy white noise as an important special case. We identify con ...

2019

Hitting with Probability One for Stochastic Heat Equations with Additive Noise

Robert Dalang, Fei Pu

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive (d-6)\documentclass[12pt]{minimal} \usepackage{ ...

Springer/Plenum Publishers2024

Critical Brownian Sheet Does Not Have Double Points

Robert Dalang

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An N-parameter Brownian sheet in R-d has double points if and only if d < 4N. In particular, in the critical case where d = 4N, the Brownian sheet does not hav ...

2012

Stochastic optimization of sailing trajectories in an upwind regatta

Sylvain Sardy, Stephan Morgenthaler, Robert Dalang

In a sailboat race, the navigator's attempts to plot the fastest possible course are hindered by shifty winds. We present mathematical models appropriate for this situation, which use statistical analysis of wind fluctuations and are amenable to stochastic ...

Palgrave Macmillan2015

Criteria for hitting probabilities with applications to systems of stochastic wave equations

Robert Dalang

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these ...

2010

Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

Robert Dalang, Fei Pu

We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of ...

ELSEVIER2021

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

Robert Dalang, Laura Claudia Catherine Vinckenbosch

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In ...

Elsevier Science Bv2014

Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension k >= 1

Robert Dalang, Fei Pu

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix gamma z of Z := (u(s, y), u(t , x) - u(s, y)), where u is the solution to a system of d non-linear stochastic heat equations in spatial dimension k >= ...

UNIV WASHINGTON, DEPT MATHEMATICS2020

Global Solutions To Stochastic Reaction-Diffusion Equations With Super-Linear Drift And Multiplicative Noise

Robert Dalang

Let xi(t, x) denote space-time white noise and consider a reaction-diffusion equation of the form ...

INST MATHEMATICAL STATISTICS2019

Second-order hyperbolic S.P.D.E.'s driven by homogeneous Gaussian noise on a hyperplane

Robert Dalang, Olivier Lévêque

We study a class of hyperbolic stochastic partial di®erential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spat ...

2006

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

Robert Dalang, Carsten Hao Ye Chong

We consider a class of parabolic stochastic PDEs on bounded domains D c Rd that includes the stochastic heat equation but with a fractional power gamma of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces ...

INT STATISTICAL INST2023

Multiple points of Gaussian random fields

Robert Dalang, Cheuk Yin Lee

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result ...

INST MATHEMATICAL STATISTICS-IMS2021

Path properties of the solution to the stochastic heat equation with Levy noise

Robert Dalang, Carsten Hao Ye Chong, Thomas Marie Jean-Baptiste Humeau

We consider sample path properties of the solution to the stochastic heat equation, in Rd or bounded domains of Rd, driven by a Levy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solut ...

SPRINGER2019

Hitting properties of parabolic S.P.D.E.'s with reflection

Robert Dalang

2006

Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

Robert Dalang, Thomas Mountford

We first consider the additive Brownian motion process (X(s(1), s(2)), (s(1), s(2)) is an element of R-2) defined by X(s(1), s(2)) = Z(1)(s(1)) - Z2(s2), where Z(1) and Z(2) are two independent (two-sided) Brownian motions. We show that with probability 1, ...

POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN2021