Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Stochastic differential equations and diffusion processes

Business Cycles: A Theoretical, Historical, and Statistical Analysis of the Capitalist Process.

We study the limit at zero of the first-passage time density of a one-dimensional diffusion process over a moving boundary and we also deal with the inverse first-passage time problem, which consists of determining the boundary shape when the first-passage density is known. Our results generalize the analogous ones already known for Brownian motion. We illustrate some examples for which the results are obtained analytically and by a numerical procedure.

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

A simple stochastic description of a model of a predator-prey system is given. The evolution of the system is described by means of Ito's stochastic differential equations (SDEs), which are the natural stochastic generalization of the Lotka-Volterra deterministic differential equations. Since these SDEs do not satisfy the usual conditions for the existence and uniqueness of the solution, we state a theorem of existence; moreover we study the stability of the equilibrium point and perform a computer simulation to study the behaviour of the trajectories of solutions with given initial data and to estimate first and second moments.

A stochastic model for predator-prey systems: basic properties, stability and computer simulation.

It is considered the integrated process $X(t)= x + \int
_0^t Y(s) ds ,$ where $Y(t)$ is a Gauss-Markov process starting
from $y.$ The first-passage time (FPT) of $X$ through a constant
boundary and the first-exit time of $X$ from an interval $(a,b)$
are investigated, generalizing some results on FPT of integrated
Brownian motion. An essential role is played by a useful
representation of $X,$
%in terms of Brownian motion
which allows
to reduces the FPT of $X$ to that of a time-changed Brownian
motion. Some explicit examples are reported; when theoretical
calculation is not available, the quantities of interest are
estimated by numerical computation.

On the first-passage time of an integrated Gauss-Markov process

We find a representation of the integral of the stationary Ornstein–Uhlenbeck (ISOU) process in terms of Brownian motion B t ; moreover, we show that, under certain conditions on the functions f and g , the double integral process (DIP) D ( t ) = ∫ β t g ( s ) ∫ α s f ( u ) d B u d s can be thought as the integral of a suitable Gauss–Markov process. Some theoretical and application details are given, among them we provide a simulation formula based on that representation by which sample paths, probability densities and first passage times of the ISOU process are obtained; the first-passage times of the DIP are also studied.

Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

We find a representation of the integral of a Gauss-Markov process in the interval [0, t], in terms of Brownian motion. Moreover, some connections with first-passagetime problems are discussed, and some examples are reported.

Mario Abundo

Papers

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

A stochastic model for predator-prey systems: basic properties, stability and computer simulation.

On the first-passage time of an integrated Gauss-Markov process

Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

On the representation of an integrated Gauss-Markov process