If a given aggregate process $S$ is a compound mixed renewal process under a probability measure $P$, we provide a characterization of all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ is converted into a compound mixed Poisson process under $Q$. This result extends earlier works of Delbaen & Haezendonck [2], Embrechts & Meister [5], Lyberopoulos & Macheras [11], and of the authors [14]. Implications to the ruin problem and to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.
Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give condit…
Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process $S$ under a probability measure $P$ we characterize all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ remains a compound renewal process under $Q$. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.
Some characterizations of mixed renewal processes in terms of exchangeability and of different types of disintegrations are given. As a consequence, an existence result for mixed renewal processes, providing also a new construction for them, is obtained. As an application, some concrete examples of constructing such processes are presented and the corresponding disintegrating measures are explicitly computed.
In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis & Macheras [24], is applied to the ruin problem in order to compute the ruin probability and to find upper and lower bounds for it.
This article is devoted to the investigation of limit theorems for mixed max-sum processes with renewal type stopping indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems.
In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from \cite{lm6z3} (enlarging Huang's \cite{hu} original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial…
Some martingale characterizations of compound mixed Poisson processes are proven, extending S. Watanabe's (1964) martingale characterization of Poisson processes as well as the main result of Lyberopoulos and Macheras (2012), concerning martingale characterizations of mixed Poisson processes.
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…
Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation princip…
If a given aggregate process $S$ is a compound mixed Poisson process under a probability measure $P$, a characterization of all probability measures $Q$ on the domain of $P$, such that $P$ and $Q$ are progressively equivalent and $S$ remains a compound mixed Poisson process with improved properties, is provided. This result generalizes earlier work of Delbaen & Haezendonck (1989). 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.
In this paper we consider the martingale Hardy spaces defined with the help of the mixed $L_{\pv}$-norm. Five mixed martingale Hardy spaces will be investigated: $H_{\pv}^{s}$, $H_{\pv}^S$, $H_{\pv}^M$, $\cP_{\pv}$ and $\cQ_{\pv}$. Several results are proved for these spaces, like atomic decompositions, Doob's inequality, boundedness, martingale inequalities and the generalization of the well-known Burkholder-Davis-Gundy inequality.
In this paper, we study optimal control problems for multiclass GI/M/n+M queues in an alternating renewal (up-down) random environment in the Halfin-Whitt regime. Assuming that the downtimes are asymptotically negligible and only the service processes are affected, we show that the limits of the diffusion-scaled state processes under non-anticipative, preemptive, work-conserving scheduling policies, are controlled jump diffusions driven by a compound Poisson jump process. We establish the asymptotic optimality of the infinite-horizon discounted and long-run average (ergodic) problems for the q…
Stacy distribution defined for the first time in 1961 provides a flexible framework for modelling of a wide range of real-life behaviours. It appears under different names in the scientific literature and contains many useful particular cases. Homogeneous Poisson processes are appropriate apriori models for the number of renewals up to a given time $t>0$. This paper mixes them and considers a Mixed Poisson process with Stacy mixing variable. We call it a Poisson-Stacy process. The resulting counting process is one of the Generalised Negative Binomial processes, and the distribution of its time…
The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida (1997) as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a m…
In [8], asymptotic expansion of the martingale with mixed normal limit was provided. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. As an application, an asymptotic expansion for a quadratic form of a diffusion process was derived in the same paper. This article gives some details of the derivation, after a short review of the martingale expansion in mixed normal limit.
In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Lévy process but by the mixed fractional Brownian motion.
A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools.
We consider many-server queueing systems with heterogeneous exponential servers and renewal arrivals. The service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy traffic limit of these queues in random environment using probability measure-valued stochastic processes. We introduce the measure-valued fairness process which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these fairness processes do not converge …
We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.
