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The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of \"disorder\" when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U, V, and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y +(t), Y −(t), and Y 0(t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y +(t) is derived. We also obtain the probability law of X(t) = Y +(t) - Y −(t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

Speeding violations are intended as both punitive and educational actions for drivers who exceed the maximum allowed speed on the roads. From a tax collection perspective, they have a significant impact on the municipal budget. Extreme Value Theory has been a valuable tool for modeling the distribution of speeding violations. Additionally, it is equally important to model the daily number of speeding occurrences. A powerful method for jointly modeling both variables is the Compound Poisson Process. By understanding both speeding behavior and the number of infractions, we can estimate the expected value of total tax collections. A mixture of Gamma densities combining with the Generalized Pareto Distribution (GPD) in tail was proposed to model the distribution of speeding values. The results indicated significant potential for tax collection. Las infracciones por exceso de velocidad están destinadas tanto a acciones punitivas como educativas para los conductores que superan la velocidad máxima permitida en las carreteras. Desde una perspectiva de recaudación fiscal, tienen un impacto significativo en el presupuesto municipal. La Teoría de Valores Extremos ha sido una herramienta valiosa para modelar la distribución de las infracciones por exceso de velocidad. Además, es igualmente importante modelar el número diario de estas infracciones. Un método poderoso para modelar ambas variables conjuntamente es el Proceso de Poisson Compuesto. Al comprender tanto el comportamiento de los conductores como el número de infracciones, podemos estimar el valor esperado de la recaudación total de impuestos. Se propuso una mezcla de densidades Gamma combinada con la Distribución Generalizada de Pareto (GPD, por sus siglas en inglés) en la cola para modelar la distribución de los valores de exceso de velocidad. Los resultados indicaron un potencial significativo para la recaudación fiscal.

Given data from a Poisson point process with intensity (x, y) ↦ n₁ (f(x) ≤ y), frequentist properties for the Bayesian reconstruction of the support boundary function f are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, (1 – α)-credible sets are also asymptotic (1 – α)-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions f outside the function class.

In this paper, we consider the composition of a homogeneous Poisson process with an independent time-fractional Poisson process. We call this composition the generalized iterated Poisson process (GIPP). The probability law in terms of the fractional Bell polynomials, governing fractional differential equations, and the compound representation of the GIPP are obtained. We give explicit expressions for mean and covariance and study the long-range dependence property of the GIPP. It is also shown that the GIPP is over-dispersed. Some results related to first-passage time distribution and the hitting probability are also examined. We define the compound and the multivariate versions of the GIPP and explore their main characteristics. Further, we consider a surplus model based on the compound version of the iterated Poisson process (IPP) and derive several results related to ruin theory. Its applications using the Poisson–Lindley and the zero-truncated geometric distributions are also provided. Finally, simulated sample paths for the IPP and the GIPP are presented.

We consider a stochastic EOQ-type model, with demand operating in a two-state random environment. This environment alternates between exponentially distributed periods of high demand and generally distributed periods of low demand. The inventory level starts at some level q, and decreases according to different compound Poisson processes during the periods of high demand and of low demand. Refilling of the inventory level to level q is required when level 0 is hit or when an expiration date is reached, whichever comes first. If such an event occurs during a high demand period, an order is instantaneously placed; otherwise, ordering is postponed until the beginning of the next high demand period. We determine various performance measures of interest, like the distribution of the inventory level at time t and of the inventory demand up to time t, the distribution of the time until refilling is required, the expected time between two refillings, the expected amount of discarded material and the expected total amount of material held in between two refillings, and the expected values of various kinds of shortages. For a given cost/revenue structure, we can thus determine the long-run average profit.

If a given aggregate process S is a mixed compound Poisson process under a probability measure P, we provide a characterization of all probability measures Q on the domain of P, such that P and Q are progressively equivalent and S remains a mixed compound Poisson process with improved properties. This result generalizes earlier work of Delbaen and Haezendonck (Insurance Math. Econom. 8 (1989) 269–277). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.

We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

In this study, we examine the theory and methodology of statistical inferences of thresholds and change-points in threshold autoregressive models. We show that least squares estimators (LSEs) of thresholds and change-points are n-consistent, and that they converge weakly to the minimizer of a compound Poisson process and the location of minima of a two-sided random walk, respectively. When the magnitude of the change in the parameters of the state regimes or in the time horizon is small, we further show that these limiting distributions can be approximated by a class of known distributions. The LSEs of the slope parameters are √n-consistent and asymptotically normal. Furthermore, a likelihood-ratio based confidence set is given for the thresholds and change-points, respectively. A Simulation study is carried out to assess the performance of our procedure, and the proposed theory and methodology are illustrated using a tree-ring data set.

We prove the existence of statistical arbitrage opportunities for jump-diffusion models of stock prices when the jump-size distribution is assumed to have finite moments. We show that to obtain statistical arbitrage, the risky asset holding must go to zero in time. Existence of statistical arbitrage is demonstrated via ‘buy-and-hold until barrier’ and ‘short until barrier’ strategies with both single and double barrier. In order to exploit statistical arbitrage opportunities, the investor needs to have a good approximation of the physical probability measure and the drift of the stochastic process for a given asset.