The paper is concerned with refining two well-known approximations to the Reed-Frost epidemic process. The first is the branching process approximation in the early stages of the epidemic; we extend its range of validity, and sharpen the estimates of the error incurred. The second is the normal...

Let Zt be a one-dimensional symmetric stable process of order [alpha] with [alpha][set membership, variant](0,2) and consider the stochastic differential equationdXt=[phi](Xt-) dZt.For [beta](1/[alpha])[logical and]1, we show there exists a function [phi] that is bounded above and below by...

In this paper we investigate the nonnegative martingale Wn=Zn/[mu]n(U), n[greater-or-equal, slanted]0 and its a.s. limit W, when (Zn)n[greater-or-equal, slanted]0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=(Un)n[greater-or-equal,...

Herein, we consider direct Markov chain approximations to the Duncan-Mortensen-Zakai equations for nonlinear filtering problems on regular, bounded domains. For clarity of presentation, we restrict our attention to reflecting diffusion signals with symmetrizable generators. Our Markov chains are...

We consider the problem of deriving the linear Boltzmann equation from the Lorentz process with hard spheres obstacles. In a suitable limit (the Boltzmann-Grad limit), it has been proved that the linear Boltzmann equation can be obtained when the position of obstacles are Poisson distributed,...

The basic coalescing random walk is a system of interacting particles. These particles start from every site of , and each moves independently as a continuous-time random walk. When two particles visit the same site, they coalesce into a single particle. We are interested in: (a) the radius...

We provide a framework for the martingale representation for futures prices which has some concrete advantages over the classical treatments of Duffie (Dynamic Asset Pricing Theory, 3rd Edition, Princeton University Press, Princeton, NJ, 2001) or Karatzas and Shreve (Brownian Motion and...

Monetary measures of risk like Value at Risk or Worst Conditional Expectation assess the risk of financial positions. The existing risk measures are of a static, one period nature. In this paper, I define dynamic monetary risk measures and I present an axiomatic approach that extends the class...

We consider a critical finite measure-valued super-Brownian motion X=(Xt,P[mu]) in , whose log-Laplace equation is associated with the semilinear equation , where the coefficient k(x)0 for the branching rate varies in space, and is continuous and bounded. Suppose that supp [mu] is compact. We...

We consider an infinite interacting particle system in which individuals choose neighbors according to evolving sets of probabilities. If x chooses y at some time, the effect is to increase the probability that y chooses x at later times. We characterize the extremal invariant measures for this...