We consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. The sampling without replacement makes it possible to take advantage of the changes in the expected value as the deck is depleted, making large bets when the game is advantageous. Using such a strategy, which is easy to implement, is known as card counting. We consider the case of a large number of decks, making an approximation to continuous time possible. A limit law of the return process is found and the optimal card counting strategy is derived. This continuous-time strategy is shown to be a natural analog of the discrete-time strategy where the so-called effects of removal are replaced by the infinitesimal generator of the card process.

The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ) being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the form a 0 / N 1/2 + a 1 / N 3/2 + · · · + b 1 / N + b 2 / N 2 + b 4 / N 4 + · · ·, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a 0, a 1, …, b 1, b 2, …}. In particular, a 0 is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremums of Lévy processes. In particular, we extend the results of Broadie, Glasserman and Kou (1999) to jump diffusion models. We also derive bounds for general exponential Lévy models.

This paper presents an analysis of a generalized version of the coupon collector problem, in which the collector receives d coupons each run and chooses the least-collected coupon so far. In the asymptotic case when the number of coupons n goes to infinity, we show that, on average, (nlogn) / d + (n / d)(m − 1)log logn + O(mn) runs are needed to collect m sets of coupons. An exact algorithm is also developed for any finite case to compute the exact mean number of runs. Numerical examples are provided to verify our theoretical predictions.

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

We study stochastic properties of the empty space for stationary germ-grain models in Rd; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

Enhanced by the global financial crisis, the discussion about an accurate estimation of regulatory (risk) capital a financial institution needs to hold in order to safeguard against unexpected losses has become highly relevant again. The presence of heavy tails in combination with small sample sizes turns estimation at such extreme quantile levels into an inherently difficult statistical issue. We discuss some of the problems and pitfalls that may arise. In particular, based on the framework of second-order extended regular variation, we compare different high-quantile estimators and propose methods for the improvement of standard methods by focusing on the concept of penultimate approximations.

A navigation on a set of points S is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where S is a nonuniform Poisson point process (in a finite domain) with intensity going to +∞. We show the convergence of the traveller's path lengths, and give the number of stages and the geometry of the traveller's trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao graphs and random θ-graphs.

Tessellations of R3 that use convex polyhedral cells to fill the space can be extremely complicated. This is especially so for tessellations which are not ‘facet-to-facet’, that is, for those where the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. Adjacency concepts between neighbouring cells (or between neighbouring cell elements) are not easily formulated when facets do not coincide. In this paper we make the first systematic study of these topological relationships when a tessellation of R3 is not facet-to-facet. The results derived can also be applied to the simpler facet-to-facet case. Our study deals with both random tessellations and deterministic ‘tilings’. Some new theory for planar tessellations is also given.