Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We propose and study a critical fluid model (or formal law of large numbers approximation) for a heavily loaded processor sharing queue. The fluid model state descriptor is a measure valued function whose dynamics are governed by a nonlinear integral equation. Under mild assumptions, we prove existence and uniqueness of fluid model solutions. Furthermore, we justify the critical fluid model as a first order approximation of a heavily loaded processor sharing queue by showing that, when appropriately rescaled, the measure valued processes corresponding to a sequence of heavily loaded processor sharing queues converge in distribution to a limit that is almost surely a fluid model solution.

The stability properties of the bandwidth allocation algorithm first fit are analyzed for the distributions concentrated on three sizes for the requests. We give the explicit expression of the ergodicity condition of this model; it involves a quadratic functional of the input parameters. The stochastic processes describing these systems are string valued Markov processes. The notion of a smooth random state is introduced. Starting from a smooth random state the fluid limits of the process can be investigated. The fluid limits of interest are random dynamical systems in $\mathbb{R}^2$ which are products of random $2\times2$ matrices.

We consider a classical risk model and allow investment into a risky asset modelled as a Black--Scholes model as well as (proportional) reinsurance. Via the Hamilton--Jacobi--Bellman approach we find a candidate for the optimal strategy and develop a numerical procedure to solve the HJB equation. We prove a verification theorem in order to show that any increasing solution to the HJB equation is bounded and solves the optimisation problem. We prove that an increasing solution to the HJB equation exists. Finally two numerical examples are discussed.

We establish the equivalence between the multivariate regular variation of a random vector and the univariate regular variation of all linear combinations of the components of such a vector. According to a classical result of Kesten [Acta Math. 131 (1973) 207-248], this result implies that stationary solutions to multivariate linear stochastic recurrence equations are regularly varying. Since GARCH processes can be embedded in such recurrence equations their finite-dimensional distributions are regularly varying.

We present a perfect simulation algorithm for stationary processes indexed by $\mathbb{Z}$, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Martínez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval.

Using the theory of set-indexed martingales, we develop a general model for survival analysis with censored data which is parameterized by sets instead of time points. We define a set-indexed Nelson--Aalen estimator for the integrated hazard function with the presence of a censoring by a random set which is a stopping set. We prove that this estimator is asymptotically unbiased and consistent. A central limit theorem is given. This model can be applied to cases when censoring occurs in geometrical objects or patterns, and is a generalization of models with multidimensional failure times.

Let $X_i$ be nonnegative, independent random variables with finite expectation, and $X^*_n = \max \{X_1, \ldots, X_n\}$. The value $EX^*_n$ is what can be obtained by a "prophet." A "mortal" on the other hand, may use $k \ge 1$ stopping rules $t_1, \ldots, t_k$, yielding a return of $E[\max_{i=1, \ldots, k} X_{t_i}]$. For $n \ge k$ the optimal return is $V^n_k (X_1, \ldots, X_n) = \sup E [\max_{i = 1, \ldots, k} X_{t_i}]$ where the supremum is over all stopping rules $t_1, \ldots, t_k$ such that $P(t_i \le n) = 1$. We show that for a sequence of constants $g_k$ which can be evaluated recursively, the inequality $EX^*_n < g_k V^n_k (X_1, \ldots, X_n)$ holds for all such $X_1, \ldots, X_n$ and all $n \ge k$; \hbox{$g_1 = 2$}, $ g_2 = 1 + e^{-1} = 1.3678\ldots,\; g_3 = 1+ e^{1-e}= 1.1793\ldots,\break g_4 = 1.0979 \ldots$ and $g_5 = 1.0567 \ldots\,$. Similar results hold for infinite sequences $X_1, X_2, \ldots\,$.

We consider a class of probabilistic models obtained by iterating random functions of $k$ random variables. We prove an analogue of the weak law of large numbers and under a symmetry condition we prove a strong law. The symmetry condition is satisfied if the initial random variables are exchangeable. Our results can be used to give stronger results than those previously obtained in the special case where the function is deterministic. Both types of models have applications in physics and in computer science.

In this work we are interested in the variations of the asymptotic shape in first passage percolation on $\mathbb{Z}^2$ according to the passage time distribution. Our main theorem extends a result proved by van den Berg and Kesten, which says that the time constant strictly decreases when the distribution of the passage time is modified in a certain manner (according to a convex order extending stochastic comparison). Van den Berg and Kesten's result requires, when the minimum $r$ of the support of the passage time distribution is strictly positive, that the mass given to $r$ is less than the critical threshold of an embedded oriented percolation model. We get rid of this assumption in the two-dimensional case, and to achieve this goal, we entirely determine the flat edge occurring when the mass given to $r$ is greater than the critical threshold, as a functional of the asymptotic speed of the supercritical embedded oriented percolation process, and we give a related upper bound for the time constant.

In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well-established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions. Our alterations also feature dramatically slower Markov chain state change rates often yielding a ten to one-hundred-fold simulation speed increase over the previous version of the method as evidenced in our computer implementations. On a weighted $L^{2}$ Hilbert space chosen to symmetrize the elliptic operator, we consider existence of and convergence to pathwise unique mild solutions of our stochastic reaction-diffusion equation. Our main convergence result, a quenched law of large numbers, establishes convergence in probability of our Markov chain approximations for each fixed path of our driving Poisson measure source. As a consequence, we also obtain the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.

We consider a general one-dimensional diffusion process and we study the probability of crossing a boundary for the associated pinned diffusion as the time at which the conditioning takes place goes to zero. We provide asymptotics for this probability as well as a first order development. We consider also the cases of two boundaries possibly depending on the time. We give applications to simulation.

The classical results on the ergodic properties of the nonlinear filter previously have been proved under the crucial assumption that the signal process and the observation noise are independent. This assumption is quite restrictive and many important problems in engineering and stochastic control correspond to filtering models with correlated signal and noise. Unlike the case of independent signal and noise, the filter process in the general correlated case may not be Markov even if the signal is a Markov process. In this work a broad class of discrete time filtering problems with signal-noise correlation is studied. It is shown that the pair process $(Y_j, \pi_j)_{j \in \mathbb{N}_0}$ is a Feller-Markov process, where $(Y_j)_{j \in \mathbb{N}_0}$ is the observation process and $\pi_j$ is the filter, that is, the conditional distribution of the signal: $X_j$ given past and current observations. It is shown that if the signal process $(X_j)$ has an invariant measure, then so does $(Y_j, \pi_j)$. Finally, it is proved that if $(X_j)$ has a unique invariant measure and the stationary flow corresponding to the signal process is purely nondeterministic, then the pair $(Y_j, \pi_j)$ has a unique invariant measure.

Given a semialgebraic set $S\subset \mathbb{R}^n$, we provide a numerical approximation procedure that provides upper and lower bounds on $\mu(S)$, for measures $\mu$ that satisfy some given moment conditions. The bounds are obtained as solutions of positive semidefinite programs that can be solved via standard software packages.