Related papers: Perpetual integral functionals of diffusions and t…
Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable function. We are interested in finding conditions on $X$ and $f$ which imply that the perpetual integral functional $$ I^X_\infty(f):=\int_0^\infty f(X_t) dt $$ is…
The distribution of the first hitting time of a disc for the standard two dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairy detailed asymptotic estimates of its density…
Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the…
By considering any one-dimensional time-homogeneous solvable diffusion process,this paper develops a complete analytical framework for computing the distribution of the last hitting time, to any level, and its joint distribution with the…
This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.
We consider exponential functionals of a multi-dimensional Brownian motion with drift, defined via a collection of linear functionals. We give a characterization of the Laplace transform of their joint law as the unique bounded solution, up…
We consider a bivariate diffusion process and we study the first passage time of one component through a boundary. We prove that its probability density is the unique solution of a new integral equation and we propose a numerical algorithm…
The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density…
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in…
Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$,…
We study the statistics of random functionals $\mathcal{Z}=\int_{0}^{\mathcal{T}}[x(t)]^{\gamma-2}dt$, where $x(t)$ is the trajectory of a one-dimensional Brownian motion with diffusion constant $D$ under the effect of a logarithmic…
This paper aims at obtaining, by means of integral transforms, analytical approximations in short times of solutions to boundary value problems for the one-dimensional reaction-diffusion equation with constant coefficients. The general form…
A popular method to compute first-passage probabilities in continuous-time Markov chains is by numerically inverting their Laplace transforms. Past decades, the scientific computing community has developed excellent numerical methods for…
S. G. Kou and H. Wang [First Passage times of a Jump Diffusion Process \textit{Ann. Appl. Probab.} {\bf 35} (2003) 504--531] give expressions of both the (real) Laplace transform of the distribution of first passage time and the (real)…
Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this…
The first-exit time process of an inverse Gaussian L\'evy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These…
This is a brief review on Brownian functionals in one dimension and their various applications, a contribution to the special issue ``The Legacy of Albert Einstein" of Current Science. After a brief description of Einstein's original…
The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in $\R^{n+1}$, composed of one-dimensional Bessel process and…
In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and difusion by suggesting and examining several Brownian functionals which characterize the lifetime and reactivity of such stochastic processes. We…
Since diffusion processes arise in so many different fields, efficient tech-nics for the simulation of sample paths, like discretization schemes, represent crucial tools in applied probability. Such methods permit to obtain approximations…