Showing papers in "Journal of Applied Probability in 2009"
TL;DR: In this article, the authors used key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality to find conditions under which the solutions to the SDEs driven by Levy noise are stable in probability.
Abstract: Using key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Levy noise are stable in probability, almost surely and moment exponentially stable.
116 citations
TL;DR: In this paper, the authors explore the behaviour of the implied volatility of a European call option far from maturity and derive asymptotic formulae with precise control over the error terms.
Abstract: This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.
74 citations
TL;DR: Albrecher and Hipp as discussed by the authors considered a Levy risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009).
Abstract: In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Levy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.
74 citations
TL;DR: In this article, the authors combine an extreme shock model with a specific cumulative shock model and derive the survival probability and the corresponding failure rate function for a system subject to a specific process of shocks.
Abstract: In extreme shock models, only the impact of the current, possibly fatal shock is usually taken into account, whereas in cumulative shock models, the impact of the preceding shocks is accumulated as well. In this paper we combine an extreme shock model with a specific cumulative shock model. It is shown that the proposed setting can also be interpreted as a generalization of the well-known Brown-Proschan model that describes repair actions for repairable systems. For a system subject to a specific process of shocks, we derive the survival probability and the corresponding failure rate function. Some meaningful interpretations and examples are discussed.
73 citations
TL;DR: In this paper, an extension of the Sparre Andersen risk model by relaxing one of its independence assumptions is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g., Assaf et al. (1984) and Kulkarni (1989)).
Abstract: In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber-Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.
72 citations
TL;DR: In this paper, a modified version of the classical optimal dividends problem of de Finetti is considered, where the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Levy process.
Abstract: We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Levy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Levy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Levy measure, the q-scale function of the spectrally negative Levy process has a derivative which is strictly log-convex.
63 citations
TL;DR: In this article, a parallel system with heterogeneous exponential component lifetimes is shown to be more skewed according to the convex transform order than the system with independent and identically distributed exponential components, and equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples are established.
Abstract: A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established A sufficient condition is also given for the proportional hazard rate model
48 citations
TL;DR: The problem of allocating k active spares to n components of a series system in order to optimize its lifetime is considered and it is shown that the strategy of balanced allocation of spares optimizes the failure rate function of the system.
Abstract: We consider the problem of allocating k active spares to n components of a series system in order to optimize its lifetime. Under the hypotheses that lifetimes of n components are identically distributed with distribution function F(-), lifetimes of k spares are identically distributed with distribution function G( ), lifetimes of components and spares are independently distributed, and that ln(G (x))/ ln(F(x)) is increasing hut, we show that the strategy of balanced allocation of spares optimizes the failure rate function of the system. Furthermore, under the hypotheses that lifetimes of n components are stochastically ordered, lifetimes of k spares are identically distributed, and that lifetimes of components and spares are independently distributed, we show that the strategy of balanced allocation of spares is superior to the strategy of allocating a larger number of components to stronger components. For coherent systems consisting of n identical components with n identical redundant (spare) components, we compare strategies of component and system redundancies under the criteria of reversed failure rate and likelihood ratio orderings. When spares and original components do not necessarily match in their life distributions, we provide a sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under reversed failure rate ordering. As a consequence, we show that, for r-out-of-rc systems, the strategy of component redundancy is superior to the strategy of system redundancy under the criterion of reversed failure rate ordering. When spares and original components match in their life distributions, we provide a necessary and sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under the likelihood ratio ordering. As a consequence, we show that, for r-out-of-n systems, with spares and original components matching in their life distributions, the strategy of component redundancy is superior to the strategy of system redundancy under the likelihood ratio ordering.
47 citations
TL;DR: It is proved that the WRMC algorithm is asymptotically better than the Metropolis–Hastings algorithm, and an estimator of the explicit optimal parameter using the proposals is proposed.
Abstract: The waste-recycling Monte Carlo (WRMC) algorithm introduced by physicists is a modification of the (multi-proposal) Metropolis-Hastings algorithm, which makes use of all the proposals in the empirical mean, whereas the standard (multi-proposal) Metropolis-Hastings algorithm uses only the accepted proposals. In this paper we extend the WRMC algorithm to a general control variate technique and exhibit the optimal choice of the control variate in terms of the asymptotic variance. We also give an example which shows that, in contradiction to the intuition of physicists, the WRMC algorithm can have an asymptotic variance larger than that of the Metropolis-Hastings algorithm. However, in the particular case of the Metropolis-Hastings algorithm called the Boltzmann algorithm, we prove that the WRMC algorithm is asymptotically better than the Metropolis-Hastings algorithm. This last property is also true for the multi proposal Metropolis-Hastings algorithm. In this last framework we consider a linear parametric generalization of WRMC, and we propose an estimator of the explicit optimal parameter using the proposals.
42 citations
TL;DR: In this article, the authors study the class of logarithmic skew-normal (LSN) distributions and show that they are all nonunique (moment-indeterminate) and explicitly describe Stieltjes classes for some LSN distributions.
Abstract: We study the class of logarithmic skew-normal (LSN) distributions. They have heavy tails; however, all their moments of positive integer orders are finite. We are interested in the problem of moments for such distributions. We show that the LSN distributions are all nonunique (moment-indeterminate). Moreover, we explicitly describe Stieltjes classes for some LSN distributions; they are families of infinitely many distributions, which are different but have the same moment sequence as a fixed LSN distribution.
40 citations
TL;DR: In this paper, the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process were analyzed in the context of a dividend barrier strategy.
Abstract: In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.
TL;DR: In this paper, the authors show how Zolotarev's duality leads to some interesting results on fractional diffusion equations employing fractional derivatives in place of the usual integer-order derivatives.
Abstract: Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Levy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1 < α < 2 to the density of the hitting time of a stable subordinator with index 1 /α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.
TL;DR: In this article, the authors developed a general method based on multivariate regular variation to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible.
Abstract: The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.
TL;DR: The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor as discussed by the authors.
Abstract: The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.
TL;DR: In this article, the authors studied several optimal stopping problems in which the gains process is a Brownian bridge or a functional function of a bridge, and they provided explicit solutions to these problems.
Abstract: We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian bridge. Our examples constitute natural finite-horizon optimal stopping problems with explicit solutions.
TL;DR: In this paper, the authors extend the martingale change-of-measure method to a class of super-diffusions and establish a Kesten-Stigum L log L type theorem for superdiffusions.
Abstract: In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).
TL;DR: In this paper, a general theorem based on relative log-concavity is shown to unify various specific results for the Poisson, binomial, negative binomial and gamma distributions in recent literature.
Abstract: We investigate stochastic comparisons between exponential family distributions and their mixtures with respect to the usual stochastic order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. A general theorem based on the notion of relative log-concavity is shown to unify various specific results for the Poisson, binomial, negative binomial, and gamma distributions in recent literature. By expressing a convolution of gamma distributions with arbitrary scale and shape parameters as a scale mixture of gamma distributions, we obtain comparison theorems concerning such convolutions that generalize some known results. Analogous results on convolutions of negative binomial distributions are also discussed.
TL;DR: In this paper, the convergence of weak and strong critical parameters of truncated branching random walks to the analogous parameters of the original branching random walk was investigated. But the convergence was not shown for the strong critical parameter of a branching random walking restricted to the cluster of a Bernoulli bond percolation.
Abstract: Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.
TL;DR: In this paper, the convergence of probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes is proved using the Wiener/Ito/Malliavin calculus.
Abstract: Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Ito/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.
TL;DR: In this article, a model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the remaining components, is proposed, motivated by the definition of sequential order statistics.
Abstract: A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.
TL;DR: In this article, the authors studied convolutions of long-tailed and subexponential distributions (probability measures), and (in passing) more general finite measures, on the real line.
Abstract: Since the inputs to such systems are frequently cumulative in their effects, the analysis of the corresponding models typically features convolutions of such heavy-tailed distributions. The properties of such convolutions depend on their satisfying certain regularity conditions. From the point of view of applications practically all such distributions may be considered to be long-tailed, and indeed to possess the stronger property of subexponentiality (see below for definitions). In this paper we study convolutions of long-tailed and subexponential distributions (probability measures), and (in passing) more general finite measures, on the real line. Our aim is to prove some important new results, and to do so through a simple, coherent and systematic approach. It turns out that all the standard properties of such convolutions are then obtained as easy consequences of these results. Thus we also hope to provide further insight into these properties, and to dispel some of the mystery which still seems to surround the phenomenon of subexponentiality in particular. Our approach is based on a simple decomposition for such convolutions, and on the concept of “h-insensitivity” for a long-tailed distribution or measure with respect to some (slowly) increasing function h. This novel approach and the basic, and very simple, new results we require are given in Section 2. In Section 3 we study convolutions of long-tailed distributions. The key results here are Theorems 5 and 6 which give conditions under which a random shifting preserves tail equivalence; in the remainder of this section we show how other (mostly known) results follow quickly and easily from our approach, and provide some generalisations. In Section 4 we similarly study convolutions of subexponential distributions. The main results here—Theorems 15 and 17—are new, as is Corollary 18; again some classical results are immediate consequences. Finally, in Section 5 we consider closure properties for the class of subexponential distributions. Theorem 20 gives a new necessary and sufficient condition for the convolution of twosubexponential distributions
TL;DR: In this paper, a random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters has been obtained and applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are discussed.
Abstract: We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.
TL;DR: In this article, the authors consider the class of Levy processes that can be written as a Brownian motion time changed by an independent Levy subordinator and prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind.
Abstract: In this paper we consider the class of Levy processes that can be written as a Brownian motion time changed by an independent Levy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.
TL;DR: In this article, the authors consider the two-dimensional version of a drainage network model and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web.
Abstract: We consider the two-dimensional version of a drainage network model introduced in Gangopadhyay, Roy and Sarkar (2004), and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed in Fontes, Isopi, Newman and Ravishankar (2002).
TL;DR: In this article, the authors proposed the binomial tree model to model the stock price dynamics and provided two rationales to support the trading strategy of "buy-and-hold for superior stock and sell-at-once for inferior stock", as suggested by conventional wisdom.
Abstract: The trading strategy of 'buy-and-hold for superior stock and sell-at-once for inferior stock', as suggested by conventional wisdom, has long been prevalent in Wall Street. In this paper, two rationales are provided to support this trading strategy from a purely mathematical standpoint. Adopting the standard binomial tree model (or CRR model for short, as first introduced in Cox, Ross and Rubinstein (1979)) to model the stock price dynamics, we look for the optimal stock selling rule(s) so as to maximize (i) the chance that an investor can sell a stock precisely at its ultimate highest price over a fixed investment horizon [0, 7]; and (ii) the expected ratio of the selling price of a stock to its ultimate highest price over [0, T]. We show that both problems have exactly the same optimal solution which can literally be interpreted as 'buy-and-hold or sell-at-once' depending on the value of p (the going-up probability of the stock price at each step): when p > \, selling the stock at the last time step N is the optimal selling strategy; when p = I, a selling time is optimal if the stock is sold either at the last time step or at the time step when the stock price reaches its running maximum price; and when p < time 0, i.e. selling the stock at once, is the unique optimal selling time.
TL;DR: In this article, a stochastic integral of Ornstein-Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution.
Abstract: In this paper, a stochastic integral of Ornstein-Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein-Uhlenbeck process is self decomposable and that the transition density is a C00-function.
TL;DR: In this article, a median of products of averages (MPA) estimator is proposed for nonreversible Markov chains on a finite state space, and sufficient conditions are given for MPA to have fixed relative precision at a given level of confidence, that is, to satisfy P(| ˆ θ − θ |≤ θe)≥ 1 − α.
Abstract: The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interest θ is represented as a product of expected values, θ = µ1 ··· µk, and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator ˆ θ, i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given for ˆ θ to have fixed relative precision at a given level of confidence, that is, to satisfy P(| ˆ θ − θ |≤ θe)≥ 1 − α. Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.
TL;DR: In this paper, the authors introduce a class of stock models that interpolates between exponential Levy models based on Brownian subordination and certain stochastic volatility models with Levy driven volatility, such as the Barndorff-Nielsen-Shephard model.
Abstract: We introduce a class of stock models that interpolates between exponential Levy models based on Brownian subordination and certain stochastic volatility models with Levy driven volatility, such as the Barndorff-Nielsen-Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Levy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.
TL;DR: In this article, the authors derived the Laplace transform of the last time before an independent, exponentially distributed time, that a spectrally negative Levy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level.
Abstract: In [5], the Laplace transform was found of the last time a spectrally negative Levy process, which drifts to innity, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Levy process drifting to infinity, is zero? In this paper we extend the result found in [5] and we derive the Laplace transform of the last time before an independent, exponentially distributed time, that a spectrally negative Levy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application we extend a result found by Doney in [6].
TL;DR: In this article, the authors generalize existing results for the steady-state distribution of growth-collapse processes and show that under certain expected conditions, point-and time-stationary versions of these processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version.
Abstract: In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Levy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.