We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

For a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at time t > 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.

Wiener integral for the coordinate process is defined under the σ-finite measure unifying Brownian penalisations, which has been introduced by [Najnudel et al., C. R. Math. Acad. Sci. Paris 345 (2007) 459–466] and [Najnudel et al., MSJ Memoirs 19. Mathematical Society of Japan, Tokyo (2009)]. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [K. Yano, J. Funct. Anal. 258 (2010) 3492–3516] of Cameron-Martin formula for the σ-finite measure.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ 0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λ n , n = 1,2,…, as well as an integral over the continuous spectrum above μ 2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ 0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

The infinitely divisible distributions on ${{\mathbb{R}}^{+}}$ of random variables ${{C}_{t}},\,{{S}_{t}}\,\text{and}\,{{T}_{t}}$ with Laplace transforms

$${{\left( \frac{1}{\cosh \sqrt{2\lambda }} \right)}^{t}},{{\left( \frac{\sqrt{2\lambda }}{\sinh \sqrt{2\lambda }} \right)}^{t}},\,\text{and}\,{{\left( \frac{\tanh \sqrt{2\lambda }}{\sqrt{2\lambda }} \right)}^{t}}$$

respectively are characterized for various $t\,>\,0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t\,=\,1\,\,\text{or}\,2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of ${{C}_{1}}\,\text{and}\,{{S}_{2}}$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from ${{S}_{t}}\,\text{and}\,{{C}_{t}}$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Let Mn denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of Mn/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

Pitman has shown that if X is Brownian motion with maximum process M, then 2M – X is a BES0(3) process. We show that this can be seen by looking at finite-dimensional densities.

A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

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.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.