Search

Sobolev type equations now constitute a vast area of non-classical equations of mathematical physics. They include equations of mathematical physics, whose representation in the form of equations or systems of partial differential equations does not fit any of the classical types (elliptic, parabolic or hyperbolic). By the properties of the operators involved, the considered Sobolev type equation has a degenerate solving semigroup of class C 0 in suitable Banach spaces. We consider a Sobolev type stochastic equation in the spaces of random processes. The concepts previously introduced for the spaces of differentiable “noises” using the Nelson–Gliklikh derivative are carried over to the case of complex-valued “noises”. We construct a solution to the weakened Showalter–Sidorov problem for Sobolev type equation with relatively p-radial operator in a space of complex-valued processes.

For Gauss-Markov processes the asymptotic behaviors of the first passage time probability density functions through certain time-varying boundaries are determined. Computational results for Wiener, Ornstein-Uhlenbeck and Brownian bridge processes show that for certain large boundaries and for large times excellent asymptotic approximations hold for such densities.

We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed., 12 pages, 4 figures

For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.

For two devices whose quality is described by non-negative one-dimensional time-homogeneous diffusion processes of the Wiener and Ornstein-Uhlenbeck types sufficient conditions are given such that their failure times, modeled as first-passage times through the zero state, are ordered according to the likelihood ratio ordering.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

Population growth is modelled by means of diffusion processes originating from fluctuation equations of a new type. These equations are obtained in the customary way by inserting random fluctuations into first order non linear differential equations. However, differently from the cases so far considered in the literature, equations possessing two non trivial fixed points are taken into account. The underlying deterministic models depict the regulated growth of a population whose size cannot decrease below some preassigned lower threshold naturally acting as an absorbing boundary. A fairly comprehensive mathematical description of these models is provided.

The recent availability of large vector computing facilities has suggest that Monte Carlo methods could be profitably used to obtain statistical information on first-passage-time problems. In the present paper the results of a number of Monte Carlo simultations performed on a CRAY X-MP/48 vector computer have been included to pinpoint precision versus computation time for evaluating mean, variance and skewness of the first-passage-time through constant and oscillatory boundaries by the Wiener and the Ornstein-Uhlenbeck processes.

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

Neuronal firing is often viewed as a first passage time problem through a time dependend boundary for a diffusion process. In this paper we approach the study of features of the first passage time probability density function for the Wiener process and for a periodic boundary. This is expected to be of interest when neuronal models include oscillatory inputs or whenever the neuron's threshold oscillates as a consequence of internal or external periodic stimulations.

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

Use of a Volterra second-kind integral equation is made to evaluate first passage time probability density functions through time varying boundaries for diffusion processes. The solutions are constructed in the form of infinite series whose terms are expressed as multidimensional integrals. An evaluation of such solutions is provided for the cases of Wiener and Ornstein-Uhlenbeck processes by standard numerical procedures, and by a Monte Carlo method. Results are discussed with reference to other existing computational methods.