The purpose of this paper is to introduce the construction of a stochastic process called "$δ$-dimensional Bessel house-moving" and its properties. We study the weak convergence of $δ$-dimensional Bessel bridges conditioned from above, and we refer to this limit as $δ$-dimensional Bessel house-moving. Applying this weak convergence result, we give the decomposition formula for its distribution and the Radon-Nikodym density for the distribution of the Bessel house-moving with respect to the one of the Bessel process. We also prove that $δ$-dimensional Bessel house-moving is a $δ$-dimensional Bessel process hitting a fixed point for the first time at $t=1$.
In this work we relate the density of the first-passage time of a Wiener process to a moving boundary with the three dimensional Bessel bridge process and a solution of the heat equation with a moving boundary. We provide bounds.
In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges obtained in a recent work by Elad Altman and Zambotti to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.
Smoothness and asymptotic behaviors are studied for the densities of the law of the occupation time on the positive line for Bessel bridges and the normalized excursion of strictly stable processes. The key role is played by these properties for functions defined by Riemann--Liouville fractional integrals.
We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge together with various relationships between the laws of these four processes.
This article shows a Bessel bridge representation for the transition density of Brownian motion on the Poincare space. This transition density is also referred to as the heat kernel on the hyperbolic space in differential geometry literature. The representation recovers the well-known closed form expression for the heat kernel on hyperbolic space in dimension three. However, the newly derived bridge representation is different from the McKean kernel in dimension two and from the Gruet's formula in higher dimensions. The methodology is also applicable to the derivation of an analogous Bessel br…
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 an SPDE 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 SPDEs associated with Bessel bridges of all dimensions smaller than 3.
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. …
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ 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>…
Given a deterministically time-changed Brownian motion $Z$ starting from 1, whose time-change $V(t)$ satisfies $V(t) > t$ for all $t > 0$, we perform an explicit construction of a process $X$ which is Brownian motion in its own filtration and that hits zero for the first time at $V(τ)$, where $τ:= \inf\{t>0: Z_t =0\}$. We also provide the semimartingale decomposition of $X$ under the filtration jointly generated by $X$ and $Z$. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process $X$ may be viewed as the analogue of a 3-dimension…
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.
In this article, we extend the integration by parts formulae for the laws of Bessel bridges obtained in previous work with Zambotti, by showing that these formulae hold for very general test functionals on $L^{2}(0,1)$. A key step consists in establishing new Taylor estimates on the laws of pinned Bessel bridges.
We generalise the integration by parts formulae obtained in arXiv:1811.00518v5 [math.PR] 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 arXiv:1811.00518v5 [math.PR] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension $2$, by providing a weak construction of the gradient dynamics corresponding to a $2$-dimensional Bessel bridge.
We obtain explicit solutions for the density $φ_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for all $t\in \mathbb{R}^+$. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge $\tilde{X}$ and exploiting its relationship with first-passage time problems as pointed out by Kardaras (2007). It turns out that this problem is related to Schrödinger's equation with time-dependent linear potential, see Feng (2001).
In this paper, we construct the Bessel line ensemble, a countable collection of continuous random curves. This line ensemble is stationary under horizontal shifts with the Bessel point process as its one-time marginal. Its finite dimensional distributions are given by the extended Bessel kernel. Furthermore, it enjoys a novel resampling invariance with respect to non-intersecting squared Bessel bridges. The Bessel line ensemble is constructed by extracting the hard edge scaling limit of a collection of independent squared Bessel processes starting at the origin and never being conditioned to i…
For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-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 prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.
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…
This paper develops some general calculus for GGC and Dirichlet process means functionals. It then proceeds via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators, to establish various distributional equivalences between these models and phenomena connected to local times and occupation times of what are defined as randomly skewed Bessel processes and bridges. This yields a host of interesting identities and explicit density formula for these models. Randomly skewed Bessel processes…
For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and ge…
Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian processes, chi-square processes, Bessel process, fractional Bessel process, Bessel bridge are examples of application of this result.
We consider Kallenberg's hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper …
Our first result concerns a characterisation by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalised version of Mecke's formula. En passant, it also allows to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump Lévy process in $\mathbb{R}^d$ with a height $a$ can be interpreted as a Poisson point process on space-time conditioned by pinning its first moment to $a$, our approach allows us to characterize bridges of Lévy processes by …
We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits a suitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and W…
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 m…
We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on $[0,1]$ pinned down in 0 at time 1 and cond…
We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.
We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and/or change of measure. All these diffusions are broadly used in mathematical finance for modelling asset prices, market indices, and interest rates. We show how the probability distributions of a squared Bessel bridge and a squared Bessel process with or withou…
This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Br…
We study the law of the minimum of a Brownian bridge, conditioned to take specific values at specific points, and the law of the location of the minimum. They are used to compare some non-adaptive optimisation algorithms for black-box functions for which the Brownian bridge is an appropriate probabilistic model and only a few points can be sampled.
It is shown that the hyperbolic functions can be associated with selfdecomposable distributions (in short: SD probability distributions or Lévy class L probability laws). Consequently, they admit associated background driving Lévy processes $Y$ (BDLP Y). We interpret the distributions of Y(1) via Bessel squared processes, Bessel bridges and local times.
This work deals with first hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distinguish two cases: (a) the case in which the process has unbounded state space before absorption, and (b) the case in which the process has bounded state space before absorption. The reason as to why this distinction has to be made will be clarified. Next, we classify processes whose local drift can be expressed as a linear combination to solut…
We prove that if the Markov generator of a diffusion process satisfies the two step strong Hörmander condition, the conditioned hypoelliptic bridge satisfies an integral bound and is a continuous semi-martingale.
Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its local time at $y$. The stochastic differential equation (SDE) whose solution corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given $a\geq 0$ is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction of the SDE the concept of {\em recurrent transformation} is introduced and {\em Bessel-type motions} as well as their SDE representations are studied. A remarkable link between an $h$-transform with a minimal excessive function and recurrent …
This paper concerns a scaling limit of a one-dimensional random walk $S^x_n$ started from $x$ on the integer lattice conditioned to avoid a non-empty finite set $A$, the random walk being assumed to be irreducible and have zero mean. Suppose the variance $σ^2$ of the increment law is finite. Given positive constants $b$, $c$ and $T$ we consider the scaled process $S^{b_N}_{[tN]}/σ\sqrt N$, $0\leq t \leq T$ started from a point $b_N \approx b\sqrt N$ conditioned to arrive at another point $\approx -c\sqrt N$ at $t=T$ and avoid $A$ in between and discuss the functional limit of it as $N\to\infty…
Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generate data with a higher sampling rate, while preserving key properties of the dynamics involved in an unbiased way. While the generation of Brownian bridges and Ornstein-Uhlenbeck bridges is well understood, unbiased generation of such stochastic bridges subject to a given extremum has been less explored in the literature. After a review of known results, we compare two algorithms for generating Brownian bridges constrained to a given extremum, one of which generalises to other diffusions. We further …
We consider a renewal process that is conditioned on the number of events in a fixed time horizon. We prove that a centered and scaled version of this process converges to a Brownian bridge, as the number of events grows large, which relies on first establishing a functional strong law of large numbers result to determine the centering. These results are consistent with the asymptotic behavior of a conditioned Poisson process. We prove the limit theorems over triangular arrays of exchangeable random variables, obtained by conditionning a sequence of independent and identically distributed rene…
It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\{S_{2n}=0\}$ the maximal displacement $\max_{k\leq 2n} |S_k|$ converges in distribution when scaled by $\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\bf{Z}$. We show that under the quenched law $P_ω$ (conditioned on the environment $ω$), the maximal displacement of the random walk when condit…
This paper deals with the rate of convergence in 1-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached 0 towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including Dirac measures and probability measures with compact support, the Wasserstein distance decays asymptotically as 1/t. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infi…
A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, …
We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer models and proves a conjecture by Caravenna and Deuschel.
For the d-dimensional model of a subcritical bond percolation (p<p_c) and a point \vec{a} in Z^d, we prove that a cluster conditioned on connecting points (0,...,0) and n\vec{a} if scaled by 1/(n|vec{a}|) along \vec{a} and by 1/sqrt{n} in the orthogonal direction converges asymptotically to Time x (d-1)-dimensional Brownian Bridge.
We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and the bottom curve are equal to Fredholm determinants whose kernel we give explicitly. In the simplest case, these determinants are expressible in terms of Painlevé V functions. We prove that as $n\to \infty$, the distributional limit of the bottom curve is the Bessel process with parameter 1/2. (This is the Bessel process associated with Dyson's Brownian motion.) We apply these results to study the expected area under the bottom and top curves.
We give a realization of the stable Lévy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to $n$. We prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the "firs…
We present a scheme for simulating conditioned semimartingales taking values in Riemannian manifolds. Extending the guided bridge proposal approach used for simulating Euclidean bridges, the scheme replaces the drift of the conditioned process with an approximation in terms of a scaled radial vector field. This handles the fact that transition densities are generally intractable on geometric spaces. We prove the validity of the scheme by a change of measure argument, and we show how the resulting guided processes can be used in importance sampling and for approximating the density of the uncon…
We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the so-called Integrated Super-Brownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive half-line. …
We consider $N$ non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large $N$ limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and to the Aztec diamond on restricted domains.
In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Kni…
A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general…
We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics…
We consider a particle system of the squared Bessel processes with index $ν> -1$ conditioned never to collide with each other, in which if $-1 < ν< 0$ the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defin…
Let $X$ be a Markov process taking values in $\mathbf{E}$ with continuous paths and transition function $(P_{s,t})$. Given a measure $μ$ on $(\mathbf{E}, \mathscr{E})$, a Markov bridge starting at $(s,\varepsilon_x)$ and ending at $(T^*,μ)$ for $T^* <\infty$ has the law of the original process starting at $x$ at time $s$ and conditioned to have law $μ$ at time $T^*$. We will consider two types of conditioning: a) {\em weak conditioning} when $μ$ is absolutely continuous with respect to $P_{s,t}(x,\cdot)$ and b) {\em strong conditioning} when $μ=\varepsilon_z$ for some $z \in \mathbf{E}$. The m…
We re-examine the celebrated Doob--McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic $α$-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic $α$-stable process, providing one interprets the components of the original identity in the right way.
