First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.      

A randomized first-passage problem for drifted Brownian motion subject to hold and jump from a boundary

We study an inverse first-passage-time problem for Wiener process X(t) subject to hold and jump from a boundary c. Let be given a threshold S > X(0) ≥ c, and a distribution function F on [0, +∞). The problem consists in finding the distribution of the holding time at c and the distribution of jumps from c, so that the first-passage time of X(t) through S has distribution F.     

Asymptotics of First-Passage Time Over a One-Sided Stochastic Boundary

We study the asymptotic behavior of the first-passage times for Brownian motion, Lévy processes and continuous martingales over one-sided increasing stochastic, as well as deterministic, boundaries. In particular, we study the first-passage time of a Brownian motion over the increasing function of its local time, give necessary and sufficient conditions for t ^–1/2 asymptotics, and obtain exact asymptotics for linear functions.     

Solving an Inverse First-Passage-Time Problem for Wiener Process Subject to Random Jumps from a Boundary

We study an inverse first-passage-time problem for Wiener process X(t) subject to random jumps from a boundary c. Let be given a threshold S > X(0); and a distribution function F on [0, + ∞). The problem consists of finding the distribution of the jumps which occur when X(t) hits c, so that the first-passage time of X(t) through S has distribution F.     

Exercisability Randomization of the American Option

Abstract

The valuation of American options is an optimal stopping time problem which typically leads to a free boundary problem. We introduce here the randomization of the exercisability of the option. This method considerably simplifies the problematic by transforming the free boundary problem into an evolution equation. This evolution equation can be transformed in a way that decomposes the value of the randomized option into a European option and the present value of continuously paid benefits. This yields a new binomial approximation for American options. We prove that the method is accurate and numerical results illustrate that it is computationally efficient.     

On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models

We consider a sample of i.i.d. times and we interpret each item as the first-passage time (FPT) of a diffusion process through a constant boundary. The problem is to estimate the parameters characterizing the underlying diffusion process through the experimentally observable FPT’s. Recently in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006) closed form estimators have been proposed for neurobiological applications. Here we study the asymptotic properties (consistency and asymptotic normality) of the class of moment type estimators for parameters of diffusion processes like those in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006). Furthermore, to make our results useful for application instances we establish upper bounds for the rate of convergence of the empirical distribution of each estimator to the normal density. Applications are also considered by means of simulated experiments in a neurobiological context.      

Parabolic Ito Equations with Mixed in Time Conditions

Abstract

We study linear stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the initial time is replaced by a condition that mixes the values of the solution at different times, including the terminal time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained.     

On nonlinear boundary value problems in the discrete setting

ABSTRACT

Results appearing in this paper can be used to establish the solvability of nonlinear discrete time systems subject to generalized nonlinear boundary conditions. Two separate sets of results are established, each of which can be used to establish existence of solutions to problems for which previous related work proves inconclusive. The first set of results imposes conditions on the sizes of nonlinearities, while the second framework requires geometric properties of the nonlinearity in the dynamics.     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.     

Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.AMS 2000 Subject Classification: 60J65, 60G40, 93E30     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Mean first-passage time in a bistable system driven by multiplicative and additive colored noises with colored cross-correlation

The effect of any system parameter (include p, q, λ, τ₁, τ₂, τ₃) on the mean first-passage time is investigated in a symmetric bistable system driven by multiplicative and additive colored noise with colored cross-correlation. Expression of the mean first-passage time (MFPT) has been obtained by applying the steepest-descent approximation. The effects of the additive noise intensity and the multiplicative noise intensity on the MFPT are mostly different only with bigger positive values of the strength of correlations between multiplicative and additive noise; and the MFPT has a maximum with the increase of the multiplicative noise intensity. The effects of the noise self-correlation times on the MFPT are quite different from that of the correlation time between the noises as the strength of correlations between multiplicative and additive noise is positive.     

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a large-deviations point of view. 2000 Mathematics Subject Classification: Primary—60K25; Secondary—60F10; 90B22     

Error Estimates for Thin Plate Spline Approximation in the Disk

This paper is concerned with approximation properties of linear combinations of scattered translates of the thin-plate spline radial basis function |·|²log|·| where the translates are taken in the unit disk D in R². We show that the L_p approximation order for this kind of approximation is 2 + 1/p (for sufficiently smooth functions), which matches Johnson's upper bound and, thus, gives the saturation order. We also show that when one increases the density of the centers at the boundary, approximation order 4 - the best possible order in the absence of a boundary - can be obtained.     

The Boundary Regularity of a Weak Solution of the Navier-Stokes Equation and its Connection to the Interior Regularity of Pressure

We assume that v is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of v near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of v.     

Fourier Method for Riemann-Hilbert-Poincaré Problem of Analytic Function

Spectrum problem with Riemann-Hilbert-Poincaré boundary condition is studied. This problem will lead to inhomogeneous Fuchsian differential equations with its right hand side depending on some constants to be determined simultaneously. We find out that the multiplicities of eigenfunctions for different eigenvalues are not necessary the same, which are in sharp contrast to the known results of Riemann-Hilbert problem for analytic functions.     

An epicycloidal boundary of the main component in the degree-n bifurcation set

It is known that the parametric boundary equation for the main component in the Mandelbrot set represents a cardioid. We derive an epicycloidal boundary equation of the main component in the degree-n bifurcation set by extending the parameter which describes the cardioid in the Mandelbrot set. Computational results as well as some useful properties are presented together with the programming source codes written inMathematica. Various boundaries are displayed for 2≤n≤7 and show a good agreement with the theory presented here. The known boundary equation enables us to significantly reduce the construction time for the degree-n bifurcation set.     

On a first-passage problem for a cumulative process with exponential decay

A first-passage problem for a cumulative process is investigated. The cumulative process is assumed to be generated by a Poisson process, and the amplitude generated by an event is assumed to decay exponentially. An integral equation for the probability density of the first-passage time until the total amplitude exceeds a pre-specified threshold level is derived. The Laplace transform of the probability density of the first-passage time is obtained explicity when each amplitude generated by an event is distributed exponentially. The mean first-passage times are given in a closed form and plotted versus the threshold level.     

First-passage times and related moments for continuous-time birth–death chains

First-passage time problems for continuous-time birth–death chains are considered. Recursive formulas for the moments of the first-exit time and of the first-passage time in terms of the potential coefficients are explicitly obtained. Making use of the probability current, some functional relations between transition probabilities for unrestricted and restricted continuous-time birth–death chains are determined. Finally, two continuous-time birth–death chains with constant rates are taken in account; for them, closed form results on the first-exit time and on the first-passage time are explicitly obtained.

     

Fast mixing for independent sets,colorings, and other models on trees

We study the mixing time of the Glauber dynamics for general spin systems on the regular tree, including the Ising model, the hard‐core model (independent sets), and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper (Martinelli, Sinclair, and Weitz, Tech. Report UCB//CSD‐03‐1256, Dept. of EECS, UC Berkeley, July 2003) in the context of the Ising model, for establishing mixing time O(nlog n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition. We also discuss applications of our framework to reconstruction problems on trees. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007