416 research outputs found
Reference data for phase diagrams of triangular and hexagonal bosonic lattices
We investigate systems of bosonic particles at zero temperature in triangular
and hexagonal optical lattice potentials in the framework of the Bose-Hubbard
model. Employing the process-chain approach, we obtain accurate values for the
boundaries between the Mott insulating phase and the superfluid phase. These
results can serve as reference data for both other approximation schemes and
upcoming experiments. Since arbitrary integer filling factors g are amenable to
our technique, we are able to monitor the behavior of the critical hopping
parameters with increasing filling. We also demonstrate that the g-dependence
of these exact parameters is described almost perfectly by a scaling relation
inferred from the mean-field approximation.Comment: 6 pages, 5 figures, accepted for publication in EP
Affine processes are regular
We show that stochastically continuous, time-homogeneous affine processes on
the canonical state space \Rplus^m \times \RR^n are always regular. In the
paper of \citet{Duffie2003} regularity was used as a crucial basic assumption.
It was left open whether this regularity condition is automatically satisfied,
for stochastically continuous affine processes. We now show that the regularity
assumption is indeed superfluous, since regularity follows from stochastic
continuity and the exponentially affine behavior of the characteristic
function. For the proof we combine classic results on the differentiability of
transformation semigroups with the method of the moving frame which has been
recently found to be useful in the theory of SPDEs
Regularity of affine processes on general state spaces
We consider a stochastically continuous, affine Markov process in the sense
of Duffie, Filipovic and Schachermayer, with cadlag paths, on a general state
space D, i.e. an arbitrary Borel subset of R^d. We show that such a process is
always regular, meaning that its Fourier-Laplace transform is differentiable in
time, with derivatives that are continuous in the transform variable. As a
consequence, we show that generalized Riccati equations and Levy-Khintchine
parameters for the process can be derived, as in the case of studied in Duffie, Filipovic and Schachermayer (2003). Moreover, we show
that when the killing rate is zero, the affine process is a semi-martingale
with absolutely continuous characteristics up to its time of explosion. Our
results generalize the results of Keller-Ressel, Schachermayer and Teichmann
(2011) for the state space and provide a new probabilistic
approach to regularity.Comment: minor correction
Affine processes are regular
We show that stochastically continuous, time-homogeneous affine processes on the canonical state space are always regular. In the paper of Duffie etal. (Ann Appl Probab 13(3):984-1053, 2003) regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine form of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDE
Polynomial processes and their applications to mathematical finance
We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo method
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