126 research outputs found
Nonexplosion criteria for relativistic diffusions
Some general Lorentz covariant operators, associated to the so-called \Theta
(or \Xi)-relativistic diffusions and making sense in any Lorentzian manifold,
have been introduced by Franchi and Le Jan [Comm. Pure Appl. Math. 60 (2007)
187-251], Franchi and Le Jan [Curvature diffusions in general relativity
(2010). Unpublished manuscript]. Only a few examples have been studied so far.
We provide in this work some nonexplosion criteria for these diffusions, which
can be used in generic cases.Comment: Published in at http://dx.doi.org/10.1214/11-AOP672 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Discounted continuous-time constrained Markov decision processes in Polish spaces
This paper is devoted to studying constrained continuous-time Markov decision
processes (MDPs) in the class of randomized policies depending on state
histories. The transition rates may be unbounded, the reward and costs are
admitted to be unbounded from above and from below, and the state and action
spaces are Polish spaces. The optimality criterion to be maximized is the
expected discounted rewards, and the constraints can be imposed on the expected
discounted costs. First, we give conditions for the nonexplosion of underlying
processes and the finiteness of the expected discounted rewards/costs. Second,
using a technique of occupation measures, we prove that the constrained
optimality of continuous-time MDPs can be transformed to an equivalent
(optimality) problem over a class of probability measures. Based on the
equivalent problem and a so-called -weak convergence of probability
measures developed in this paper, we show the existence of a constrained
optimal policy. Third, by providing a linear programming formulation of the
equivalent problem, we show the solvability of constrained optimal policies.
Finally, we use two computable examples to illustrate our main results.Comment: Published in at http://dx.doi.org/10.1214/10-AAP749 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Exponential martingales and changes of measure for counting processes
We give sufficient criteria for the Dol\'eans-Dade exponential of a
stochastic integral with respect to a counting process local martingale to be a
true martingale. The criteria are adapted particularly to the case of counting
processes and are sufficiently weak to be useful and verifiable, as we
illustrate by several examples. In particular, the criteria allow for the
construction of for example nonexplosive Hawkes processes as well as counting
processes with stochastic intensities depending on diffusion processes
Large deviations and a Kramers' type law for self-stabilizing diffusions
We investigate exit times from domains of attraction for the motion of a
self-stabilized particle traveling in a geometric (potential type) landscape
and perturbed by Brownian noise of small amplitude. Self-stabilization is the
effect of including an ensemble-average attraction in addition to the usual
state-dependent drift, where the particle is supposed to be suspended in a
large population of identical ones. A Kramers' type law for the particle's exit
from the potential's domains of attraction and a large deviations principle for
the self-stabilizing diffusion are proved. It turns out that the exit law for
the self-stabilizing diffusion coincides with the exit law of a potential
diffusion without self-stabilization and a drift component perturbed by average
attraction. We show that self-stabilization may substantially delay the exit
from domains of attraction, and that the exit location may be completely
different.Comment: Published in at http://dx.doi.org/10.1214/07-AAP489 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
PDEs for the joint distributions of the Dyson, Airy and Sine processes
In a celebrated paper, Dyson shows that the spectrum of an n\times n random
Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves
as n noncolliding Brownian motions held together by a drift term. The universal
edge and bulk scalings for Hermitian random matrices, applied to the Dyson
process, lead to the Airy and Sine processes. In particular, the Airy process
is a continuous stationary process, describing the motion of the outermost
particle of the Dyson Brownian motion, when the number of particles gets large,
with space and time appropriately rescaled. In this paper, we answer a question
posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy
process at two different times. Similarly we find a PDE satisfied by the joint
distribution of the Sine process. This hinges on finding a PDE for the joint
distribution of the Dyson process, which itself is based on the joint
probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE
for the Dyson process is then subjected to an asymptotic analysis, consistent
with the edge and bulk rescalings. The PDEs enable one to compute the
asymptotic behavior of the joint distribution and the correlation for these
processes at different times t_1 and t_2, when t_2-t_1\to \infty, as
illustrated in this paper for the Airy process. This paper also contains a
rigorous proof that the extended Hermite kernel, governing the joint
probabilities for the Dyson process, converges to the extended Airy and Sine
kernels after the appropriate rescalings.Comment: Published at http://dx.doi.org/10.1214/009117905000000107 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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