Renewal series and square-root boundaries for Bessel processes

In this paper, we present a switching property for squared Bessel process. We prove that the starting point and the time parameter can be in a sense switched. As a consequence, we obtain new distributional dependencies among a squared Bessel process R, a Brownian motion with drift $$ {B}^{\left(\mu \right)},\mu \ge 0,\left({B}_t^{\left(\mu \right)}:= {B}_t+\mu t\kern0.5em \mathrm{for}\ \mathrm{a}\ \mathrm{Brownian}\ \mathrm{motion}\ B\right), $$ and an integral functional of geometric Brownian motion A(μ) connected by the Lamperti relation. Finally, we deduce a new explicit form of the joint distribution of the Lamperti triple $$ \left({R}_T,{A}_t^{\left(\mu \right)},\exp \left(2{B}_t^{\left(\mu \right)}\right)\right) $$ for fixed t > 0 andT > 0.

Asymmetric skew Bessel processes and their applications to finance

Filtering of absorbing and reflecting Brownian motions

This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite dimensional space S'R endowed with the gaussian measure mu. The Hilbert space ( L2) is defined as (L2) (L2) ≡ L2( S'R , mu) and our results are based on the Gel'fand triple ( S )beta ⊂ (L2) ⊂ S* b . The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In Chapter 3 we present a generalization of the Ito Formula to anticipating processes in the white noise framework. We first introduce an extension of the Ito integral to anticipating processes called the Hitsuda Skorokhod integral. We then use the anticipating Ito formula for processes of the type theta(X(t), B(c)), where X(t) is a Wiener integral and B(t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion for a ≤ c ≤ b, which was obtained in Professor H. H. Kuo's book, White Noise Distribution Theory, as a motivation to obtain our first main result. We generalize the formula in Prof. Kuo's book to processes of the form theta(X(t),F) where X(t) is a Wiener integral and F ∈ W 1/2, a special subspace of (L 2). Chapter 4 contains the second part of our work. We first state the Clark-Ocone (C-O) formula in the white noise framework. We then verify the formula for Brownian functionals using the very important white noise tool, the S-transform. As our second main result, we extend the formula to generalized Brownian functionals of the form F=Sinfinityn=0 :˙⊗n:,fn where fn ∈ L 2 R ⊗&d4;n . As examples we verify the formula for the Hermite Brownian functional: B(t)n:t using the Ito formula. We also compute the C-O formula for the Donsker's delta function using the result for the Hermite Brownian functional. Finally, the formula is generalized to compositions of tempered distributions with Brownian motion, i.e, to functionals of the form f( B(t)) where f ∈ S'R and B(t) = B( t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion.

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary t ↦ a + bt, a ≥ 0, b ∈ ℝ, by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter δ > 0 and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

Covariance and independence, rotational symmetry, isonormal Gaussian process, independent increments, Brownian motion and bridge, scaling and inversion, Gaussian Markov processes, quadratic variation, path irregularity, strong Markov and reflection properties, Bessel processes, maximum process, arcsine and uniform laws, laws of the iterated logarithm, Wiener integral, spectral and moving-average representations, Ornstein–Uhlenbeck process, multiple Wiener–Ito integrals, chaos expansion of variables and processes.

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 orderν> −1 with constant driftμ≠ 0. Whenν> -½ andμ< 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 <ν< -½ andμ< 0, there is only one eigenvalueλ0= 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 goal of this chapter is to give a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The latter appears in the pricing of Asian options. Bessel processes are defined and some of their properties are given. The known expressions for the probability density function of the integral of geometric Brownian motion are stated, and other related results are given, in particular the Geman and Yor (1993) Laplace transform for Asian option prices.

A necessary and sufficient condition is given for a Brownian excursion law in a Lipschitz domain to share the local path properties with an excursion law in a halfspace. This condition is satisfied for all boundary points of every C1't-domain, ct > 0. There exists a Cl-domain such that the condition is satisfied almost nowhere on the boundary. A probabilistic interpretation and applications to minimal thinness and boundary behavior of Green functions are given. 0. Introduction. We will show that the local path properties of Brownian excursions in Cl,--domains, a > 0, are the same as the local path properties of Brownian excursions in a half-space (Theorem 3.1). This need not be the case if the domain is of class C' (Proposition 3.1). The method of proof is based on the exit system theory of Maisonneuve [12]. We will study Brownian excursion laws, i.e., one of the ingredients of an exit system. We will give a necessary and sufficient condition for an excursion law in a Lipschitz domain to have the same local path properties as an excursion law in a half-space (Theorems 2.1 and 2.2). This last result can be applied to obtain new criteria for minimal thinness (Theorem 4.1) and existence of a nondegenerate normal derivative for the Green function (Theorem 4.2). Local path properties of 1-dimensional Brownian excursions have been known for some time. They are the same as the local path properties of the 3-dimensional Bessel process (see ?2.10 of Ito and McKean [11] or ?II.67 of Williams [21]). Dvoretsky and Erd6s [8] and Shiga and Watanabe [18] have given Kolmogorov-type tests for Bessel processes. Local path properties of excursions of multidimensional Brownian motion have been studied by Shimura [19] (excursions of 2-dimensional Brownian motion from a line) and Burdzy [1] (n-dimensional case, n > 2). Our theorem about minimal thinness generalizes results of Essen and Jackson [9] and Burdzy [3, 4]. A new criterion for the existence of the nondegenerate normal derivative for the Green function extends some results of Widman [20], Rippon [17] and Burdzy [4]. The present article mainly uses ideas from excursion theory (see Maisonneuve [12], Williams [21] and Burdzy [1]) and potential theory (see Doob [7]). Received by the editors July 26, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 60G17, 60J50, 60J65; Secondary 31B25, 60J45.

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.

The limitations of traditional indifference-zone (IZ) ranking and selection procedures stem from their reliance on an IZ parameter, which is often unknown in practice and can lead to inefficiencies. Additionally, since exact evaluation of the probability of correct selection (PCS) is generally unattainable, these approaches commonly rely on a Bonferroni-Inequality-type lower bound to ensure the worst-case PCS, which introduces further conservativeness. To overcome these challenges, we propose fully sequential IZ-free procedures that eliminate the need for an IZ parameter and reduce the conservativeness associated with Bonferroni Inequality approaches. We demonstrate that, in the context of IZ-free procedures, the screening statistic corresponds to the Euclidean distance of a multi-dimensional standard Brownian motion, which behaves as a Bessel process. This connection simplifies the derivation of statistical guarantees for the proposed procedures and facilitates the analytical expressions for key parameters. Our procedures accommodate both known equal variances and unknown unequal variances, with statistical guarantees established in the known equal variance case. Numerical experiments confirm that the proposed procedures outperform existing methods in reducing PCS conservativeness and improving sample size efficiency. These advancements enhance the robustness and practicality of IZ-free procedures for diverse applications.

This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.

In this paper, we derive exact closed-form density functions of the generalized Verhulst process and the Bessel process with constant drift, which have applications in mathematical biology and queueing theory. We propose a generic probabilistic method for deriving exact closed-form density functions for these two diffusion processes based on a novel application of the exponential measure change in [T. Hurd and A. Kuznetsov, Explicit formulas for Laplace transforms of stochastic integrals, Markov Process. Relat. Fields, 14(2):277–290, 2008], together with formulae in [A. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, Birkhauser, Basel, 2015]. Our study generalizes several known results in the literature.

In this paper, we will present a strong (or pathwise) approximation of\nstandard Brownian motion by a class of orthogonal polynomials. The coefficients\nthat are obtained from the expansion of Brownian motion in this polynomial\nbasis are independent Gaussian random variables. Therefore it is practical\n(requires $N$ independent Gaussian coefficients) to generate an approximate\nsample path of Brownian motion that respects integration of polynomials with\ndegree less than $N$. Moreover, since these orthogonal polynomials appear\nnaturally as eigenfunctions of an integral operator defined by the Brownian\nbridge covariance function, the proposed approximation is optimal in a certain\nweighted $L^{2}(\\mathbb{P})$ sense. In addition, discretizing Brownian paths as\npiecewise parabolas gives a locally higher order numerical method for\nstochastic differential equations (SDEs) when compared to the standard\npiecewise linear approach. We shall demonstrate these ideas by simulating\nInhomogeneous Geometric Brownian Motion (IGBM). This numerical example will\nalso illustrate the deficiencies of the piecewise parabola approximation when\ncompared to a new version of the asymptotically efficient log-ODE (or\nCastell-Gaines) method.\n

Pitman transforms and Brownian motion in the interval viewed as an affine alcove