Relativistic analysis of stochastic kinematics

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient $D$ measured in a frame $\Sigma$ moving with velocity $w$ with respect to the rest frame of the stochastic process can be expressed as $D= D_0 \, \gamma^{-3}(w)$. Subsequently, higher dimensional processes are analyzed, and it is shown that the diffusivity tensor in a moving frame becomes non-isotropic with $D_\parallel = D_0 \, \gamma^{-3}(w)$, and $D_\perp = D_0 \, \gamma^{-1}(w)$, where $D_\parallel$ and $D_\perp$ are the diffusivities parallel and orthogonal to the velocity of the moving frame. The analysis of discrete Space-Time Diffusion processes permits to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed

Spatial birth-and-death processes with a finite number of particles

Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over $\mathbb{R} ^ \mathrm{d}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. We also study pathwise behavior of an aggregation model. The probability of extinction and the growth rate of the number of particles conditioning on non-extinction are estimated.Comment: arXiv admin note: substantial text overlap with arXiv:1502.06783. New version note: significant structural and other change

Interest rate models with Markov chains

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Faà di Bruno’s formula and spatial cluster modelling

AbstractThe probability generating functional (p.g.fl.) provides a useful means of compactly representing point process models. Cluster processes can be described through the composition of p.g.fl.s, and factorial moment measures and Janossy measures can be recovered from the p.g.fl. using variational derivatives. This article describes the application of a recent result in variational calculus, a generalisation of Faà di Bruno’s formula, to determine such results for cluster processes

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog

Regulation mechanisms in spatial stochastic development models

The aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models. We describe the density behavior for models with global mortality and local establishment rates. We prove that the local self-regulation via a competition mechanism (density dependent mortality) may suppress a unbounded growth of the averaged density if the competition kernel is superstable.Comment: 19 page