715 research outputs found
Feller property and infinitesimal generator of the exploration process
We consider the exploration process associated to the continuous random tree
(CRT) built using a Levy process with no negative jumps. This process has been
studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is
a useful tool to study CRT as well as super-Brownian motion with general
branching mechanism. In this paper we prove this process is Feller, and we
compute its infinitesimal generator on exponential functionals and give the
corresponding martingale
Multiplicative decompositions and frequency of vanishing of nonnegative submartingales
In this paper, we establish a multiplicative decomposition formula for
nonnegative local martingales and use it to characterize the set of continuous
local submartingales Y of the form Y=N+A, where the measure dA is carried by
the set of zeros of Y. In particular, we shall see that in the set of all local
submartingales with the same martingale part in the multiplicative
decomposition, these submartingales are the smallest ones. We also study some
integrability questions in the multiplicative decomposition and interpret the
notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio
Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure
Many transport processes in nature exhibit anomalous diffusive properties
with non-trivial scaling of the mean square displacement, e.g., diffusion of
cells or of biomolecules inside the cell nucleus, where typically a crossover
between different scaling regimes appears over time. Here, we investigate a
class of anomalous diffusion processes that is able to capture such complex
dynamics by virtue of a general waiting time distribution. We obtain a complete
characterization of such generalized anomalous processes, including their
functionals and multi-point structure, using a representation in terms of a
normal diffusive process plus a stochastic time change. In particular, we
derive analytical closed form expressions for the two-point correlation
functions, which can be readily compared with experimental data.Comment: Accepted in Phys. Rev. Let
The role of the agent's outside options in principal-agent relationships
We consider a principal-agent model of adverse selection where, in order to trade with the principal,
the agent must undertake a relationship-specific investment which affects his outside option to trade,
i.e. the payoff that he can obtain by trading with an alternative principal. This creates a distinction
between the agentâs ex ante (before investment) and ex post (after investment) outside options to trade.
We investigate the consequences of this distinction, and show that whenever an agentâs ex ante and ex
post outside options differ, this may equip the principal with an additional tool for screening among
different agent types, by randomizing over the probability with which trade occurs once the agent
has undertaken the investment. In turn, this may enhance the efficiency of the optimal second-best
contract
Bessel processes, the Brownian snake and super-Brownian motion
We prove that, both for the Brownian snake and for super-Brownian motion in
dimension one, the historical path corresponding to the minimal spatial
position is a Bessel process of dimension -5. We also discuss a spine
decomposition for the Brownian snake conditioned on the minimizing path.Comment: Submitted to the special volume of S\'eminaire de Probabilit\'es in
memory of Marc Yo
Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum
Motivated by a problem in climate dynamics, we investigate the solution of a
Bessel-like process with negative constant drift, described by a Fokker-Planck
equation with a potential V(x) = - [b \ln(x) + a\, x], for b>0 and a<0. The
problem belongs to a family of Fokker-Planck equations with logarithmic
potentials closely related to the Bessel process, that has been extensively
studied for its applications in physics, biology and finance. The Bessel-like
process we consider can be solved by seeking solutions through an expansion
into a complete set of eigenfunctions. The associated imaginary-time
Schroedinger equation exhibits a mix of discrete and continuous eigenvalue
spectra, corresponding to the quantum Coulomb potential describing the bound
states of the hydrogen atom. We present a technique to evaluate the
normalization factor of the continuous spectrum of eigenfunctions that relies
solely upon their asymptotic behavior. We demonstrate the technique by solving
the Brownian motion problem and the Bessel process both with a negative
constant drift. We conclude with a comparison with other analytical methods and
with numerical solutions.Comment: 21 pages, 8 figure
Bessel bridges decomposition with varying dimension. Applications to finance
We consider a class of stochastic processes containing the classical and
well-studied class of Squared Bessel processes. Our model, however, allows the
dimension be a function of the time. We first give some classical results in a
larger context where a time-varying drift term can be added. Then in the
non-drifted case we extend many results already proven in the case of classical
Bessel processes to our context. Our deepest result is a decomposition of the
Bridge process associated to this generalized squared Bessel process, much
similar to the much celebrated result of J. Pitman and M. Yor. On a more
practical point of view, we give a methodology to compute the Laplace transform
of additive functionals of our process and the associated bridge. This permits
in particular to get directly access to the joint distribution of the value at
t of the process and its integral. We finally give some financial applications
to illustrate the panel of applications of our results
Stochastic integration based on simple, symmetric random walks
A new approach to stochastic integration is described, which is based on an
a.s. pathwise approximation of the integrator by simple, symmetric random
walks. Hopefully, this method is didactically more advantageous, more
transparent, and technically less demanding than other existing ones. In a
large part of the theory one has a.s. uniform convergence on compacts. In
particular, it gives a.s. convergence for the stochastic integral of a finite
variation function of the integrator, which is not c\`adl\`ag in general.Comment: 16 pages, some typos correcte
Dual random fragmentation and coagulation and an application to the genealogy of Yule processes
The purpose of this work is to describe a duality between a fragmentation
associated to certain Dirichlet distributions and a natural random coagulation.
The dual fragmentation and coalescent chains arising in this setting appear in
the description of the genealogy of Yule processes.Comment: 14 page
Bridge Decomposition of Restriction Measures
Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding
walks in the upper half plane, we show that the conjectured scaling limit of
the half-plane SAW, the SLE(8/3) process, also has an appropriately defined
bridge decomposition. This continuum decomposition turns out to entirely be a
consequence of the restriction property of SLE(8/3), and as a result can be
generalized to the wider class of restriction measures. Specifically we show
that the restriction hulls with index less than one can be decomposed into a
Poisson Point Process of irreducible bridges in a way that is similar to Ito's
excursion decomposition of a Brownian motion according to its zeros.Comment: 24 pages, 2 figures. Final version incorporates minor revisions
suggested by the referee, to appear in Jour. Stat. Phy
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