We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.
We are concerned with the first hitting times of the Bessel processes. We give explicit expressions for the densities by means of the zeros of the Bessel functions and show their asymptotic behavior.
We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover all the cases.
The asymptotic behavior of the tail probabilities for the first hitting times of the Bessel process with arbitrary index is shown without using the explicit expressions for the distribution function obtained in the authors' previous works.
In this paper we pursue and complete the study of the simulation of the hitting time of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work, the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method was based mainly on the explicit formula for the distribution of the hitting time and on the connexion between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non inte…
In this note we shortly recall the importance of last-hitting times in theory and applications of optimal stopping. As a small contribution to this domain we then propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from r > 0 at its ultimate minimum. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the forthcoming reading guide of Nikeghbali and Platen on this subject.
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>…
We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{é}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.
In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that e…
Let $T_1^{(μ)}$ be the first hitting time of the point 1 by the Bessel process with index $μ\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(μ)}(t)$ of $T_1^{(μ)}$, obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of $T_1^{(μ)}$ which exibit the dependence both on time and space variables. Our result provides optimal estimates for the density of the hitting time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve existing bounds. Another application is to provide sharp estimates for the …
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $δ$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $δ$, provided $δ\le 0$. As an application, we show that the SLE($κ$) welding homeomorphism is continuous in $κ$ for $κ\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_κ$ in $κ$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
A new stochastic process is introduced and considered - squared Bessel process with special stochastic time. The analogues of fundamental properties for Brownian motion are deduced for squared Bessel process. In particular an analogue of the celebrated strong Markov construction of Brownian motion independent of a given sigma field is presented and proved. This result has strong consequences. It allows for deeper understanding the nature of first hitting time of squared Bessel process. The joint distribution of two correlated squared Bessel processes is presented. For squared Bessel process it…
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…
We recover in part a recent result of Hamana-Matsumoto (2014) on the asymptotic behaviors for tail probabilities of first hitting times of Bessel process. Our proof is based on a weak convergence argument. The same reasoning enables us to derive the asymptotic behaviors for the tail probability of the time at which the global infimum of Bessel process is attained, and for expected values relative to local infima. In addition, we give another proof of the result of Hamana-Matsumoto with improvement of error estimates, which complements in the case of noninteger dimensions the asymptotic formula…
We look at decompositions of perpetuities and apply that to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by David M. DeLong and M. Yor. Several random factorizations and characterizations of the studied distributions are established.
We consider a Brownian motion forced to stay in the quadrant by an electrostatic oblique repulsion from the sides. We tackle the question of hitting the corner or an edge, and find product-form stationary measures under a certain condition, which is reminiscent of the skew-symmetry condition for an obliquely reflected Brownian motion.
We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $γ$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.
In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using aga…
We consider a $d$-dimensional random field $u=(u(x), x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D\subset \mathbb{R}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from [Dalang and Sanz-Solé, 2009], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on $L^2$ estimates of inc…
In this paper we introduce a new method for the simulation of the exit time and position of a $δ$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $δ$-dimensional Bessel process and the $δ$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with exis…
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, …
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
I consider a stochastic optimization problem for a time-changed Bessel process whose diffusion rate is constrained to be between two positive values $r_{1}<r_{2}$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equat…
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.
We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -δ/2x + o(1/x) with δ> -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}ϕ(n), where c = (3+δ)/2 and ϕis a slowly varying function given explicitly in terms of the o(…
Let $X$ be a squared Bessel process. Following a Feynman-Kac approach, the Laplace transforms of joint laws of $(U, \int_0^{R_y}X_s^p\,ds)$ are studied where $R_y$ is the first hitting time of $y$ by $X$ and $U$ is a random variable measurable with respect to the history of $X$ until $R_y$. A subset of these results are then used to solve the associated small ball problems for $\int_0^{R_y}X_s^p\,ds$ and determine a Chung's law of iterated logarithm. $(\int_0^{R_y}X_s^p\,ds)$ is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The fi…
This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probabil…
In this paper, we derive an integral representation for the density of the reciprocal of the first hitting time of the boundary of a wedge of angle $π/4$ by a radial Dunkl process with equal multiplicity values. Not only this representation readily yields the non negativity of the density, but also provides an analogue of Dufresne's result on the distribution of the first hitting time of zero by a Bessel process and a generalization of the Vakeroudis-Yor's identity satisfied by the first exit time from a wedge by a planar Brownian motion. We also use a result due to Spitzer on the angular part…
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 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 consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.
We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_N$ these Bessel processes are related with $β$-Hermite and $β$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko-Pastu…
The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best tool for problems connected with the Kendall generalized convolution.
In the last years, many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling li…
The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear whether the exit time can be approximated by hitting times of approximations of the process. We prove a functional limit theorem for approximating weakly both the paths of the Markov process and its exit times. In contrast to the functional limit theorem in [arXiv:1902.06249v1] for approximating the paths, we impose a stronger assumption here. This is essential, as we present an example showing that the theorem extende…
We study Boolean stable laws, $\mathbf{b}_{α,ρ}$, with stability index $α$ and asymmetry parameter $ρ$. We show that the classical scale mixture of $\mathbf{b}_{α,ρ}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{α,ρ}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{α,ρ}$ is both classically and freely infinitely divisible for $α\leq1/2$ and also for some $α>1/2$. Furthermore, we show the multiplicative i…
In the recent paper \cite{Ng5} we have introduced a method of studying the multi-dimensional Kingman convolutions and their associated stochastic processes by embedding them into some multi-dimensional ordinary convolutions which allows to study multi-dimensional Bessel processes in terms of the cooresponding Brownian motions. Our further aim in this paper is to introduce k-dimensional Kingman-Lévy (KL) processes and prove some of their fluctuation properties which are analoguous to that of k-symmetric Lévy processes. In particular, the Lévy-Itô decomposition and the series representation of R…
For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $ν_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.
We prove estimates at infinity of convolutions $f^{n\star}$ and densities of the corresponding compound Poisson measures for a class of radial decreasing densities on $\mathbb{R}^d$, $d \geq 1$, which are not convolution equivalent. Existing methods and tools are limited to the situation in which the convolution $f^{2\star}(x)$ is comparable to initial density $f(x)$ at infinity. We propose a new approach which allows one to break this barrier. We focus on densities which are products of exponential functions and smaller order terms -- they are common in applications. In the case when the smal…
