Bessel Bridges Research Articles

Construction and sample path properties of Brownian house-moving between two curves

AbstractThis study aims to construct a stochastic process called “Brownian house-moving,” which is a Brownian bridge conditioned to stay between two curves. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motions, conditioned Brownian bridges, and conditioned three-dimensional Bessel bridges. Moreover, the sample path properties of Brownian house-moving are studied as well.

Taylor estimates for the laws of pinned Bessel bridges, and integration by parts

Taylor estimates for the laws of pinned Bessel bridges, and integration by parts

Series Expansions and Direct Inversion for the Heston Model

Efficient sampling for the conditional time integrated variance process in the Heston stochastic volatility model is key to the simulation of the stock price based on its exact distribution. We construct a new series expansion for this integral in terms of double infinite weighted sums of particular independent random variables through a change of measure and the decomposition of squared Bessel bridges. When approximated by series truncations, this representation has exponentially decaying truncation errors. We propose feasible strategies to largely reduce the implementation of the new series to simulations of simple random variables that are independent of any model parameters. We further develop direct inversion algorithms to generate samples for such random variables based on Chebyshev polynomial approximations for their inverse distribution functions. These approximations can be used under any market conditions. Thus, we establish a strong, efficient, and almost exact sampling scheme for the Heston model.

Bessel SPDEs with general Dirichlet boundary conditions

We generalise the integration by parts formulae obtained in a recent work with Zambotti to Bessel bridges on [0,1] with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of [7] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension 2, by providing a construction of the stationary dynamics corresponding to the law of a 2-dimensional Bessel bridge.

Integration by parts formulae for the laws of Bessel bridges via hypergeometric functions

In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges recently obtained in [2] to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.

Bessel SPDEs and renormalised local times

In this article, we prove integration by parts formulae (IbPFs) for the laws of Bessel bridges from 0 to 0 over the interval [0, 1] of dimension smaller than 3. As an application, we construct a weak version of a stochastic PDE having the law of a one-dimensional Bessel bridge (i.e. the law of a reflected Brownian bridge) as reversible measure, the dimension 1 being particularly relevant in view of applications to scaling limits of dynamical critical pinning models. We also exploit the IbPFs to conjecture the structure of the stochastic PDEs associated with Bessel bridges of all dimensions smaller than 3.

On an approach to boundary crossing by stochastic processes

In this paper we provide an overview as well as new (definitive) results of an approach to boundary crossing. The first published results in this direction appeared in de la Peña and Giné (1999) book on decoupling. They include order of magnitude bounds for the first hitting time of the norm of continuous Banach-Space valued processes with independent increments. One of our main results is a sharp lower bound for the first hitting time of càdlàg real-valued processes X(t), where X(0)=0 with arbitrary dependence structure: ETrγ≥∫01{a−1(rα)}γdα, where Tr=inf{t>0:X(t)≥r},a(t)=E{sup0≤s≤tX(s)} and γ>0. Under certain extra conditions, we also obtain an upper bound for ETrγ. As the main text suggests, although Tr is defined as the hitting time of X(t) hitting a level boundary, the bounds developed can be extended to more general processes and boundaries. We shall illustrate applications of the bounds derived for additive processes, Gaussian Processes, Bessel Processes, Bessel bridges among others. By considering the non-random function a(t), we can show that in various situations, ETr≈a−1(r).

On hitting times, Bessel bridges and Schrödinger’s equation

In this paper, we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schrödinger’s equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov’s theorem, which suggests a strategy to extend our ideas to problems in $\mathbb{R}^{n}$ and general diffusions. This approach also leads to (c) because we may relate, through a Feynman–Kac representation, functionals of a Bessel bridge with two Schrödinger-type problems. In particular, we also find a fundamental solution to a class of parabolic partial differential equations with linear potential. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.

Bessel Bridges Decomposition with Varying Dimension: Applications to Finance

We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.

On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

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.

The double CFTP method

We consider the problem of the exact simulation of random variables Z that satisfy the distributional identity Z = L VY + (1- V ) Z , where V ∈ [0,1] and Y are independent, and = L denotes equality in distribution. Equivalently, Z is the limit of a Markov chain driven by that map. We give an algorithm that can be automated under the condition that we have a source capable of generating independent copies of Y , and that V has a density that can be evaluated in a black-box format. The method uses a doubling trick for inducing coalescence in coupling from the past. Applications include exact samplers for many Dirichlet means, some two-parameter Poisson--Dirichlet means, and a host of other distributions related to occupation times of Bessel bridges that can be described by stochastic fixed point equations.

Lamperti-type laws

This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an α-stable law, for 0 < α < 1, and the other variable has the law defined by polynomially tilting the density of an α-stable random variable by a factor θ > −α. When θ = 0, these variables equate with the ratio investigated by Lamperti [Trans. Amer. Math. Soc. 88 (1958) 380–387] which, remarkably, was shown to have a simple density. This variable arises in a variety of areas and gains importance from a close connection to the stable laws. This rationale, and connection to the PD (α, θ) distribution, motivates the investigations of its generalizations which we refer to as Lamperti-type laws. We identify and exploit links to random variables that commonly appear in a variety of applications. Namely Linnik, generalized Pareto and z-distributions. In each case we obtain new results that are of potential interest. As some highlights, we then use these results to (i) obtain integral representations and other identities for a class of generalized Mittag–Leffler functions, (ii) identify explicitly the Lévy density of the semigroup of stable continuous state branching processes (CSBP) and hence corresponding limiting distributions derived in Slack and in Zolotarev [Z. Wahrsch. Verw. Gebiete 9 (1968) 139–145, Teor. Veroyatn. Primen. 2 (1957) 256–266], which are related to the recent work by Berestycki, Berestycki and Schweinsberg, and Bertoin and LeGall [Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 214–238, Illinois J. Math. 50 (2006) 147–181] on beta coalescents. (iii) We obtain explicit results for the occupation time of generalized Bessel bridges and some interesting stochastic equations for PD (α, θ)-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension 2 − 2α.

Gamma expansion of the Heston stochastic volatility model

We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the Pitman–Yor decomposition of Bessel bridges. We combine this representation with the Broadie–Kaya exact simulation method and use it to circumvent the most time-consuming step in that method.

Maximum distributions of bridges of noncolliding Brownian paths

One-dimensional Brownian motion starting from the origin at time t=0 , conditioned to return to the origin at time t=1 and to stay positive during time interval 0<t<1, is called the Bessel bridge with duration 1. We consider an N-particle system of such Bessel bridges conditioned never to collide with each other in 0<t<1, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum values of paths attained in the time interval t epsilon (0,1) are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general N. We show that the present N-path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the 2Nx2N matrix-valued Brownian bridge in the symmetry class C. Using this fact, computer simulations are performed and numerical results on the N dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related to random matrix theory, the representation theory of symmetry, and number theory.

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.

Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions

Smoothness and asymptotic behaviors of the probability densities of the occupation time are studied for Bessel bridges and stable excursions. The difficulty is that the densities are implicitly expressed using Riemann–Liouville fractional integrals.

Gamma Expansion of the Heston Stochastic Volatility Model

We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the Pitman-Yor decomposition of Bessel bridges. We combine this representation with the Broadie-Kaya exact simulation method and use it to circumvent the most time-consuming step in that method.

An interpretation and some generalizations of the Anderson–Darling statistics in terms of squared Bessel bridges

Some generalizations of the Anderson–Darling statistics are presented, as well as their representations as functions of Bessel processes. Combining these results with some computations of Donati–Martin–Yor (Ann. Probab. 25(3) (1997) 1011–1058), standard results are recovered.

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X t (δ) ,t≥0) be the BESQδ process starting at δx. We are interested in large deviations as ${{\delta \rightarrow \infty}}$ for the family {δ−1 X t (δ) ,t≤T}δ, – or, more generally, for the family of squared radial OUδ process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramer–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.

Integration by parts on Bessel Bridges and related stochastic partial differential equations

We prove integration by parts formulae with respect to the law of Bessel Bridges of dimension δ⩾3. For δ=3 we have an infinite-dimensional boundary measure, and for δ>3 a singular logarithmic derivative. We give applications to SPDEs with additive space-time white noise and singular drifts, whose solutions are non-negative. To cite this article: L. Zambotti, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 209–212.