518,023 research outputs found
From discrete to continuous time evolutionary finance
This paper aims to open a new avenue for research in continuous-time financial market models with endogenous prices and heterogenous investors. To this end we introduce a discrete-time evolutionary stock market model that accommodates time periods of arbitrary length. The dynamics is time-consistent and allows the comparison of paths with different frequency of trade. The main result in this paper is the derivation of the limit model as the length of the time period tends to zero. The resulting model in continuous time generalizes the workhorse model of mathematical finance by introducing asset prices that are driven by the market interaction of investors following self-financing trading strategies. Our approach also offers a numerical scheme for the simulation of the continuous-time model that satisfies constraints such as market clearing at every time step. An illustration is provided
From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution
The discrete time quantum walk defined as a quantum-mechanical analogue of
the discrete time random walk have recently been attracted from various and
interdisciplinary fields. In this review, the weak limit theorem, that is, the
asymptotic behavior, of the one-dimensional discrete time quantum walk is
analytically shown. From the limit distribution of the discrete time quantum
walk, the discrete time quantum walk can be taken as the quantum dynamical
simulator of some physical systems.Comment: This is the invited review paper on the special issue on Theoretical
and Mathematical Aspects of Discrete Time Quantum Walk from Journal of
Computational and Theoretical Nanoscienc
Transition from discrete to continuous time of arrival distribution for a quantum particle
We show that the Kijowski distribution for time of arrivals in the entire
real line is the limiting distribution of the time of arrival distribution in a
confining box as its length increases to infinity. The dynamics of the confined
time of arrival eigenfunctions is also numerically investigated and
demonstrated that the eigenfunctions evolve to have point supports at the
arrival point at their respective eigenvalues in the limit of arbitrarilly
large confining lengths, giving insight into the ideal physical content of the
Kijowsky distribution.Comment: Accepted for publication in Phys. Rev.
Crossovers induced by discrete-time quantum walks
We consider crossovers with respect to the weak convergence theorems from a
discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk
(CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs
in some limit. At first we generalize our previous study [Phys. Rev. A
\textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show
that the position measurements per each step with probability can be evaluated, where is the final time and . We
also give a corresponding continuous-time case. As a consequence, crossovers
from the diffusive spreading (random walk) to the ballistic spreading (quantum
walk) can be seen as the parameter shifts from 0 to 1 in both discrete-
and continuous-time cases of the weak convergence theorems. Secondly, we
introduce a new class of the DTQW, in which the absolute value of the diagonal
parts of the quantum coin is proportional to a power of the inverse of the
final time . This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW
is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence
theorem for the FTD-DTQW which shows a variety of spreading properties.
Finally, we consider the FTD-DTQW with periodic position measurements. This
weak convergence theorem gives a phase diagram which maps sufficiently
long-time behaviors of the discrete- and continuous-time quantum and random
walks.Comment: 14 pages, 1 figur
Phase limitations of Zames-Falb multipliers
Phase limitations of both continuous-time and discrete-time Zames-Falb
multipliers and their relation with the Kalman conjecture are analysed. A phase
limitation for continuous-time multipliers given by Megretski is generalised
and its applicability is clarified; its relation to the Kalman conjecture is
illustrated with a classical example from the literature. It is demonstrated
that there exist fourth-order plants where the existence of a suitable
Zames-Falb multiplier can be discarded and for which simulations show unstable
behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is
developed. Its application is demonstrated with a second-order counterexample
to the Kalman conjecture. Finally, the discrete-time limitation is used to show
that there can be no direct counterpart of the off-axis circle criterion in the
discrete-time domain
Discrete and continuous time simulations of spatial ecological processes predict different final population sizes and interspecific competition outcomes
Cellular automata (CAs) are commonly used to simulate spatial processes in ecology. Although appropriate for modelling events that occur at discrete time points, they are also routinely used to model biological processes that take place continuously. We report on a study comparing predictions of discrete time CA models to those of their continuous time counterpart. Specifically, we investigate how the decision to model time discretely or continuously affects predictions regarding long-run population sizes, the probability of extinction and interspecific competition. We show effects on predicted ecological outcomes, finding quantitative differences in all cases and in the case of interspecific competition, additional qualitative differences in predictions regarding species dominance. Our findings demonstrate that qualitative conclusions drawn from spatial simulations can be critically dependent on the decision to model time discretely or continuously. Contrary to our expectations, simulating in continuous time did not incur a heavy computational penalty. We also raise ecological questions on the relative benefits of reproductive strategies that take place in discrete and continuous time
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