A new analytical approximation tool, derived from the classical PDEs’ theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives, and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Mor...

Given a Markov chain with uncertain transition probabilities modelled in a Bayesian way, we investigate a technique for analytically approximating the mean transition frequency counts over a finite horizon. Conventional techniques for addressing this problem either require the enumeration of a set of generalized process "hyperstates" whose cardinality grows exponentially with the terminal horiz...

A Poisson equation in d for the elliptic operator corresponding to an ergodic diffusion process is considered. Existence and uniqueness of its solution in Sobolev classes of functions is established along with the bounds for its growth. This result is used to study a diffusion approximation for two-scaled diffusion processes using the method of corrector; the solution of a Poisson equation serv...

We consider the branching and annihilating random walk A → 2A and 2A → 0 with reaction rates σ and λ, respectively, and hopping rate D, and study the phase diagram in the (λ/D, σ/D) plane. According to standard mean-field theory, this system is in an active state for all σ/D > 0, and perturbative renormalization suggests that this mean-field result is valid for d > 2; however, nonperturbative r...

Confinement of cosmic rays by turbulent magnetic fields and the relation between magnetic and cosmic ray energy densities are investigated. Cosmic rays are assumed to diffuse preferentially along magnetic field lines. The implementation into a nonconservative high-order finite difference code is discussed. By solving an explicitly time-dependent equation for the cosmic ray energy flux, causalit...

We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explicit. We provide non asymptotic error bounds, whic...

This review surveys a significant set of recent ideas developed in the study of nonlinear Galerkin approximation. A significant role is played by the Krasnosel’skii Calculus, which represents a generalization of the classical inf-sup linear saddle point theory. A description of a proper extension of this calculus, and the relation to the inf-sup theory are part of this review. The general study...

Understanding variation in allele frequencies across populations is a central goal of population genetics. Classical models for the distribution of allele frequencies, using forward simulation, coalescent theory, or the diffusion approximation, have been applied extensively for demographic inference, medical study design, and evolutionary studies. Here we propose a tractable model of ordinary d...

In the paper a stochastic control model is studied, that admits a diffusion approximation. In the prelimit model the disturbances are given by noise processes of various types: additive stationary noise, rapidly oscillating processes, and discontinuous processes with large intensity for jumps of small size. We show that a feedback control, that satisfies a Lipschitz condition and is δ−optimal f...

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.