We provide a general framework for dual representations of Laplace transforms of Markov processes. Such representations state that the Laplace transform of a finite-dimensional distribution of a Markov process can be expressed in terms of a Laplace transform involving another Markov process, but with coefficients in the Laplace transform and time indices of the process interchanged. Dual representations of Laplace transforms have been used recently to study open ASEP and to describe stationary measures of the open KPZ equation. Our framework covers both recently discovered examples in the literature and several new ones, involving general Lévy processes and certain birth-and-death processes.
We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this wo…
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).
For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain.
We study distributions of meeting times for finite symmetric Markov chains. For Markov kernels defined on large state spaces which satisfy certain weak inhomogeneity in return probabilities of points up to large numbers of steps, we obtain approximation, with explicit error bounds, of the Laplace transforms of some meeting times (without scaling) by ratios of Green functions closely related to hitting times of points. In studying this approximation, we identify some key matrix power series in Markov kernels weighted with solutions to a discrete transport-like equation with explicit coefficient…
The Laplace transform of the $d$-dimensional distribution of Brownian excursion is expressed as the Laplace transform of the $(d+1)$-dimensional distribution of an auxiliary Markov process, started from a $σ$-finite measure and with the roles of arguments and times interchanged. A similar identity holds for the Laplace transform of a generalized meander, which is expressed as the Laplace transform of the same auxiliary Markov process, with a different initial law.
Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitiona…
The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a…
The probabilistic interpretation of Laplace transforms is used to help to describe the Laplace Transform $L(s)$ of improper random variables. In particular, busy periods in queueing models are examined. The value of $L(0)$ is explained in certain special cases.
The Laplace transform is a widely used tool in the study of probability distributions, often allowing for a probability density functions and distribution functions simpler determination and being a moments generating function. In this paper it is considered a situation not so simple, as it is the case of the MGinf queue busy period length distribution. Attention will also be given the respective tail Laplace transform. Then, in the context of an open queues network, which nodes behave as MGinf queues, the Laplace transform will be used to construct an algorithm to determine the Laplace transf…
We investigate a refracted Levy process driven by a jump diffusion process, whose jumps have rational Laplace transforms. For such a stochastic process, formulas for the Laplace transform of its occupation times are deduced. To derive the main results, some modifications on our previous approach have been made. In addition, we obtain a very interesting identity, which is conjectured to hold for a general refracted Levy process.
In this article, we considers reversible Markov chains of which $L^2$-distances can be expressed in terms of Laplace transforms. The cutoff of Laplace transforms was first discussed by Chen and Saloff-Coste in [8], while we provide here a completely different pathway to analyze the $L^2$-distance. Consequently, we obtain several considerably simplified criteria and this allows us to proceed advanced theoretical studies, including the comparison of cutoffs between discrete time lazy chains and continuous time chains. For an illustration, we consider product chains, a rather complicated model wh…
We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulas for the Laplace transform of its distribution. Our formulas are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. Besides, the results are used to price Variable Annuities with state-dependent fees.
We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the form of a Mellin-Barnes integral. As a corollary, we obtain an integral expression for the characteristic function; we show that the integral expression derived by Leipnik in [11] is incorrect. We present two approximations for the Laplace transform of the lognormal distribution, both valid in $\mathbb{C} \setminus(-\infty,0]$. In the last section, we discuss …
We consider the random variables $R$ which are solutions of the distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent of $R$ and $\ABS{M}\leq 1$. Goldie and Grübel showed that the tails of $R$ are no heavier than exponential. In this note we provide the exact lower and upper bounds of the domain of the Laplace transform of $R$.
The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
We prove sharp estimates on heat kernels and Green functions for subordinate Markov processes with both discrete an continuous time, under relatively weak assumptions about original processes as well as Laplace exponents of subordinators. We also show stability of heat kernel estimates in the discrete settings.
We study the following question: if $f$ is a nonzero measurable function on $[0,\infty)$ and $m$ and $n$ distinct nonnegative integers, does the ratio $\widehat{f^n}/\widehat{f^m}$ of the Laplace transforms of the powers $f^n$ and $f^m$ of $f$ uniquely determine $f$? The answer is yes if one of $m, n$ is zero, by the inverse Laplace transform. Under some assumptions on the smoothness of $f$ we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory where $f$ is the cumulative distribution function of a certain…
We present the idea of intertwining of two diffusions by Feynman-Kac operators. We present some variations and implications of the method and give examples of its applications. Among others, it turns out to be a very useful tool for finding the expectations of some functionals of diffusions, especially for computing the Laplace transforms of stochastic processes. The examples give new results on marginal distributions of many stochastic processes including a generalized squared Bessel processes and joint distribution for squared Bessel bridge and its integral - the close formulae of the Laplac…
A popular method to compute first-passage probabilities in continuous-time Markov chains is by numerically inverting their Laplace transforms. Past decades, the scientific computing community has developed excellent numerical methods for solving problems governed by partial differential equations (PDEs), making the availability of a Laplace transform not necessary here for computational purposes. In this study we demonstrate that numerical PDE problem solvers are suitable for computing first passage times, and can be very efficient for this purpose. By doing extensive computational experiments…
In this paper, we will give a sufficient condition for a non-negative random variable $X$ to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.
It is well known that the Laplace-Stieltjes transform of a nonnegative random variable (or random vector) uniquely determines its distribution function. We extend this uniqueness theorem by using the Muntz-Szasz Theorem and the identity for the Laplace-Stieltjes and Laplace-Carson transforms of a distribution function. The latter appears for the first time to the best of our knowledge. In particular, if X and Y are two nonnegative random variables with joint distribution H, then H can be characterized by a suitable set of countably many values of its bivariate Laplace-Stieltjes transform. The …
The 1993 Laplace transform approach of Geman and Yor is a celebrated advance in valuing Asian options. Its insights are fundamental from both a mathematical and a financial perspective. In this paper, we discuss two observations regarding the financial relevance of its results. First, we show that the Geman and Yor Laplace transform is not that of an Asian option price, as reported in Geman and Yor and other papers. We nonetheless show how the Geman and Yor Laplace transform can be used to obtain the price of an Asian option. Second, we find that following Geman and Yor these Laplace transfoms…
We calculate exactly the Laplace transform of the Fréchet distribution in the form $γx^{-(1+γ)} \exp(-x^{-γ})$, $γ> 0$, $0 \leq x < \infty$, for arbitrary rational values of the shape parameter $γ$, i.e. for $γ= l/k$ with $l, k = 1,2, \ldots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fréchet distribution serves as a kernel of Fréchet integral transform. It turns out that the Fréchet transform of one-sided Lévy law reproduces the Fréchet distribution.
The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform, inversion formulae for the Laplace transform are of central importance; such formulae are old and well-known (Fourier-Mellin or Bromwich integral, Post-Widder inversion). The present work is motivated from an elementary statistical problem, namely, the unbiased estimation of a parametric function of the scale in the basic model of a random sample from exponential…
We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.
This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that is described by the Itô's stochastic ordinary differential equation of fractional Brownian motion. Furthermore, this paper applies the derived transition probability density function to the Cox-Ingersoll-Ross model governed by the fractional Brownian motion instead of the usual Br…
