413 research outputs found
Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed semimartingale
is a time-changed stochastic integral driven by the original semimartingale. As
a direct consequence, a specialized form of the Ito formula is derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with drift
extend to a larger class of stochastic differential equations involving a
time-change with continuous paths. A form of the general solution of linear
equations in this new class is established, followed by consideration of some
examples analogous to the classical equations. Through these examples, each
coefficient of the stochastic differential equations in the new class is given
meaning. The new feature is the coexistence of a usual drift term along with a
term related to the time-change.Comment: 27 pages; typos correcte
A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise
We prove a functional non-central limit theorem for jump-diffusions with
periodic coefficients driven by strictly stable Levy-processes with stability
index bigger than one. The limit process turns out to be a strictly stable Levy
process with an averaged jump-measure. Unlike in the situation where the
diffusion is driven by Brownian motion, there is no drift related enhancement
of diffusivity.Comment: Accepted to Journal of Theoretical Probabilit
Fractional smoothness and applications in finance
This overview article concerns the notion of fractional smoothness of random
variables of the form , where is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages
Relativistic diffusion processes and random walk models
The nonrelativistic standard model for a continuous, one-parameter diffusion
process in position space is the Wiener process. As well-known, the Gaussian
transition probability density function (PDF) of this process is in conflict
with special relativity, as it permits particles to propagate faster than the
speed of light. A frequently considered alternative is provided by the
telegraph equation, whose solutions avoid superluminal propagation speeds but
suffer from singular (non-continuous) diffusion fronts on the light cone, which
are unlikely to exist for massive particles. It is therefore advisable to
explore other alternatives as well. In this paper, a generalized Wiener process
is proposed that is continuous, avoids superluminal propagation, and reduces to
the standard Wiener process in the non-relativistic limit. The corresponding
relativistic diffusion propagator is obtained directly from the nonrelativistic
Wiener propagator, by rewriting the latter in terms of an integral over
actions. The resulting relativistic process is non-Markovian, in accordance
with the known fact that nontrivial continuous, relativistic Markov processes
in position space cannot exist. Hence, the proposed process defines a
consistent relativistic diffusion model for massive particles and provides a
viable alternative to the solutions of the telegraph equation.Comment: v3: final, shortened version to appear in Phys. Rev.
Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond
We present a simple construction method for Feller processes and a framework
for the generation of sample paths of Feller processes. The construction is
based on state space dependent mixing of L\'evy processes.
Brownian Motion is one of the most frequently used continuous time Markov
processes in applications. In recent years also L\'evy processes, of which
Brownian Motion is a special case, have become increasingly popular.
L\'evy processes are spatially homogeneous, but empirical data often suggest
the use of spatially inhomogeneous processes. Thus it seems necessary to go to
the next level of generalization: Feller processes. These include L\'evy
processes and in particular Brownian motion as special cases but allow spatial
inhomogeneities.
Many properties of Feller processes are known, but proving the very existence
is, in general, very technical. Moreover, an applicable framework for the
generation of sample paths of a Feller process was missing. We explain, with
practitioners in mind, how to overcome both of these obstacles. In particular
our simulation technique allows to apply Monte Carlo methods to Feller
processes.Comment: 22 pages, including 4 figures and 8 pages of source code for the
generation of sample paths of Feller processe
A stochastic network with mobile users in heavy traffic
We consider a stochastic network with mobile users in a heavy-traffic regime.
We derive the scaling limit of the multi-dimensional queue length process and
prove a form of spatial state space collapse. The proof exploits a recent
result by Lambert and Simatos which provides a general principle to establish
scaling limits of regenerative processes based on the convergence of their
excursions. We also prove weak convergence of the sequences of stationary joint
queue length distributions and stationary sojourn times.Comment: Final version accepted for publication in Queueing Systems, Theory
and Application
Last passage percolation and traveling fronts
We consider a system of N particles with a stochastic dynamics introduced by
Brunet and Derrida. The particles can be interpreted as last passage times in
directed percolation on {1,...,N} of mean-field type. The particles remain
grouped and move like a traveling wave, subject to discretization and driven by
a random noise. As N increases, we obtain estimates for the speed of the front
and its profile, for different laws of the driving noise. The Gumbel
distribution plays a central role for the particle jumps, and we show that the
scaling limit is a L\'evy process in this case. The case of bounded jumps
yields a completely different behavior
Convergence of repeated quantum non-demolition measurements and wave function collapse
Motivated by recent experiments on quantum trapped fields, we give a rigorous
proof that repeated indirect quantum non-demolition (QND) measurements converge
to the collapse of the wave function as predicted by the postulates of quantum
mechanics for direct measurements. We also relate the rate of convergence
toward the collapsed wave function to the relative entropy of each indirect
measurement, a result which makes contact with information theory.Comment: 4 pages, 1 figur
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