The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems of birth-and-death type in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [Kur73]. Our approach is based on the study of forward Kolmogorov eq...

Abstract Stationary distributions of Markov processes can typically be characterized as probability measures that annihilate the generator in the sense that ∫ E Afdμ = 0 for f ∈ D(A); that is, for each such μ, there exists a stationary solution of the martingale problem for A with marginal distribution μ. This result is extended to models corresponding to martingale problems that include absolu...

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic di erential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an \ordinary" sense over an arbitrarily prescribed time duration, via a direct \Four Step Scheme". Using this scheme, we further p...

In this paper we investigate the nature of the adapted ;solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an "ordinary" sense over an arbitrarily prescribed time duration, via a direct "Four Step Scheme". Using this scheme, we further...

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fr...

We consider the averaging principle for stochastic reaction–diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical ...

in a real separable Hilbert space H driven by an infinite dimensional cylindrical symmetric Lévy process Z = (Zt). The process Z may take values in a Hilbert space 1 Supported by the M.I.U.R. research projects Prin 2004 and 2006 “Kolmogorov equations” and by the Polish Ministry of Science and Education project 1PO 3A 034 29 “Stochastic evolution equations with Lévy noise”. 2 Supported by the Po...

We present an automated formulation mechanism that facilitates the inclusion of batches of geometrically distributed size in Markov modulated, multiprocessor queues of finite or infinite capacity. This provides a practical approach for the analytical modelling of many present day communication and computer systems, e.g. the Internet and mobile networks. Geometric distributions can be scaled and...

A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space L(H, ν) where ν is an infinitesimally excessive measure. Then a weak solution is constructed. 2000 Mathematics Subject Classification AMS: 76S05, 35J25, 37L40.

Monte Carlo estimators of sensitivity indices and the marginal density of the price dynamics are derived for the Hobson-Rogers stochastic volatility model. Our approach is based mainly upon the Kolmogorov backward equation by making full use of the Markovian property of the dynamics given the past information. Some numerical examples are presented with a GARCHlike volatility function and its ex...