Let be a Poisson process of intensity one in the infinite plane ℝ2. We surround each point x of by the open disc of radius r centred at x. Now let Sn be a fixed disc of area n, and let Cr(Sn) be the set of discs which intersect Sn. Write Erk for the event that Cr(Sn) is a k-cover of Sn, and Frk for the event that Cr(Sn) may be partitioned into k disjoint single covers of Sn. We prove that P(Erk ∖ Frk) ≤ ck / logn, and that this result is best possible. We also give improved estimates for P(Erk). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of ℝ2 with half-planes that cannot be partitioned into two single covers.

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤd and is given by a deterministic function of the coin flips. Let ZΦ be the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour of ZΦ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε < ∞ for every ε > 0.

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ON-OFF input process. We study stochastic control problems associated with the long-run average cost, the infinite-horizon discounted cost, and the finite-horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the long-run average cost problem. We also establish Abelian limit relationships among the value functions of the above control problems.

In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.