Saddlepoint approximations are developed for Markov random walks $S_n$ and are used to evaluate the probability that $(j-i) g((S_j - S_i)/(j-i))$ exceeds a threshold value for certain sets of $(i,j)$. The special case $g(x) = x$ reduces to the usual scan statistic in change-point detection problems, and many generalized likelihood ratio detection schemes are also of this form with suitably chosen $g$. We make use of this boundary crossing probability to derive both the asymptotic Gumbel-type distribution of scan statistics and the asymptotic exponential distribution of the waiting time to false alarm in sequential change-point detection. Combining these saddlepoint approximations with truncation arguments and geometric integration theory also yields asymptotic formulas for other nonlinear boundary crossing probabilities of Markov random walks satisfying certain minorization conditions.

Logarithmic asymptotics are proved for the tail of the supremum of a stochastic process, under the assumption that the process satisfies a restricted large deviation principle on regularly varying scales. The formula for the rate of decay of the tail of the supremum, in terms of the underlying rate function, agrees with that stated by Duffield and O'Connell [Math. Proc. Cambridge Philos. Soc. (1995) 118 363-374]. The rate function of the process is not assumed to be convex. A number of queueing examples are presented which include applications to Gaussian processes and Weibull sojourn sources.

We consider a discrete-time queue with general service distribution and characterize a class of arrival processes that possess a large deviation rate function that remains unchanged in passing through the queue. This invariant rate function corresponds to a kind of exponential tilting of the service distribution. We establish a large deviations analogue of quasireversibility for this class of arrival processes. Finally, we prove the existence of stationary point processes that have a probability law that is preserved by the queueing operator and conjecture that they have large deviation rate functions which belong to the class of invariant rate functions described above.

Let $ \{Z_{n}, n \ge 1 \}$ be a single type supercritical Galton--Watson process with mean $EZ_{1} \equiv m$, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence $\{R_n \equiv \frac{Z_{n+1}}{Z_n}\dvtx n \ge 1 \}$ and establishes a "phase transition" in rates depending on whether $r$, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant $\alpha$. This is done via a careful analysis of the harmonic moments of $Z_n$.

For a multitype branching process in varying environment convergent in probability, a certain sequence of linear combinations of the type sizes is shown to possess some convergence properties. This sequence turns out to be instrumental in deriving a condition for continuity of the limiting distribution function. An application to an $L_1$ convergent process whose offspring mean matrices are weakly ergodic is also given.

Super Brownian motion is known to occur as the limit of properly rescaled interacting particle systems such as branching random walk, the contact process and the voter model.

In this paper we show that certain linearly interacting diffusions converge to super Brownian motion if suitably rescaled in time and space. The results comprise nearest neighbor interaction as well as long range interaction.

This article deals with a way to solve the spatially homogeneous Landau equation using probabilistic tools. Thanks to the study of a nonlinear stochastic differential equation driven by a space-time white noise, we state the existence of a measure solution of the Landau equation with probability measure initial data, for a generalization of the Maxwellian molecules case. Then, by approximation of the Landau coefficients, the first result helps us to state the existence of a measure solution for some soft potentials [$\gamma \in ( -1,0) $]. This second part is based on the use of nonlinear stochastic differential equations and some martingale problems.

We introduce a new interacting particle system to investigate the behavior of the nonlinear, nonlocal diffusive equation already studied by Benachour et al. [3, 4]. We first prove an uniform (with respect to time) propagation of chaos. Then, we show that the solution of the nonlinear PDE converges exponentially fast to equilibrium recovering a result established by an other way by Carrillo, McCann and Vilanni [7]. At last we provide explicit and Gaussian confidence intervals for the convergence of an implicit Euler scheme to the stationary distribution of the nonlinear equation.

Loss networks in heavy traffic under Kelly's scaling are analyzed. In the case of a single node and $R$ classes of calls, a degenerate diffusion approximation theorem around the corresponding fluid limit in an $({R-1})$-dimensional hyperplane is proved.

Consider a fluid queue with a finite buffer $B$ and capacity $c$ fed by a superposition of $N$ independent On--Off processes. An On--Off process consists of a sequence of alternating independent periods of activity and silence. Successive periods of activity, as well as silence, are identically distributed. The process is active with probability $p$ and during its activity period produces fluid at constant rate $r$. For this queueing system, under the assumption that the excess activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary overflow probability and loss rate. In the case of homogeneous processes with excess activity periods equal in distribution to $\tau^e$, the queue loss rate is asymptotically, as $B\rightarrow \infty$, equal to \[ \Lambda^B = (r_0 -c) \pmatrix{N\cr m} \Bigl(p \,\Pr\Bigl[\tau^e > \frac{B}{r_0-c}\Bigr]\Bigr)^{m} (1+o(1)), \] where $m$ is the smallest integer greater than $(c-N\rho)/(r-\rho)$, $r_0= m r + (N -m)\rho$, $\rho=r p$ and $N \rho < c$; the results require a mild technical assumption that $(c-N\rho)/(r-\rho)$ is not an integer. The analyzed queueing system represents a standard model of resource sharing in telecommunication networks. The derived asymptotic results are shown to provide accurate approximations to simulation experiments. Furthermore, the results offer insight into qualitative tradeoffs between the overflow probability, offered traffic load, capacity and buffer space.

Algebraic convergence in the $L^2$-sense is studied for general time-continuous, reversible Markov chains with countable state space, and especially for birth--death chains. Some criteria for the convergence are presented. The results are effective since the convergence region can be completely covered, as illustrated by two examples.

We define the family of transient Markov arrival processes (transient MAPs) which combine features of transient (or terminating) renewal processes and of the well-known MAPs: transient MAPs are point processes on the line, controlled by a finite Markov chain, which almost surely comprise a finite number of points. We analyze their basic properties.

For a LARCH ("linear ARCH") sequence $ (y_n, \sigma_n) $ exhibiting long range dependence, we determine the limiting distribution of sums $\sum f(y_n)$, $ \sum f(\sigma_n) $ for smooth functions $ f $ satisfying $E(y_0 f' (y_0)) \neq 0 $, $ E (\sigma_0 f' (\sigma_0)) \neq 0 $. We also give an approximation formula for the above sums, providing the first term of the asymptotic expansions of $ \sum f (y_n),\break \sum f (\sigma_n)$.

Partial immunization processes are generalizations of the contact process in which the susceptibility of a site to infection depends on whether or not it has been previously infected. Such processes can exhibit a phase of weak survival, in which the process survives but drifts off to infinity, even on graphs such as $\mathbb{Z}^d$, where no such phase exists for the contact process. We establish that whether or not strong survival occurs depends only on the rate at which sites are reinfected and not on the rate at which sites are infected for the first time. This confirms a prediction by Grassberger, Chaté and Rousseau. We then study the processes on homogeneous trees, where the behaviour is also related to that of the contact process whose infection rate is equal to the reinfection rate of the partial immunization process. However, the phase diagram turns out to be substantially richer than that of either the contact process on a tree or partial immunization processes on $\mathbb{Z}^d$.

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with $N$ particles in a rectangle of $\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{+ 1}$, where $q\in (0,1)$ is a measure of the drift driving the particles toward the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in $N$ and in the size of the rectangle. The proof is inspired by a recent interesting technique envisioned by E. Carlen, M. C. Carvalho and M. Loss to analyze the Kac model for the nonlinear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin-$S$, $S\in \ov2\bbN$, $\mbox{\textit{{XXZ}}}$ chain and for the 111 interface of the spin-$S$ $\mbox{\textit{{XXZ}}}$\vspace{-1pt} Heisenberg model, thus generalizing previous results valid only for spin $\ov2$.

Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius $r$, have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems.

We look at the percolation properties of this generalized model, showing that an unbounded connected component of discs does not exist, almost surely, for small values of the density $\lambda$ of the Poisson point process, for any covering algorithm. In general, it turns out not to be true that unbounded connected components arise when $\lambda$ is taken sufficiently high. However, we identify some large families of covering algorithms, for which such an unbounded component does arise for large values of $\lambda$.

We show how a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of $\lambda$, for any covering algorithm.

Finally, we show that a large class of covering algorithms, which arise in many practical applications, can get arbitrarily close to achieving a minimal density of covering discs. We also construct an algorithm that achieves this minimal density.

We study the minimal investment that is needed in order to super-replicate (i.e., hedge with certainty) continuous-time options under transaction costs. We deal with both exotic and path-independent European and American options. In all our examples we prove that the optimal strategy is the cheapest possible buy and hold. Our method is to study the problem in a discrete-time shadow market that is free of transaction costs where the options are perpetual. We also produce useful and precise estimates of potential capital gains in a transaction cost environment. We believe that our method is robust and has both theoretical and practical implications. One advantage of our approach, in contrast with the existing literature, is that we do not impose any trading strategies restrictions related to the no bankruptcy condition. Namely we allow hedging with unlimited borrowing and still the best one can do is buy and hold. Another advantage is that we do not assume that share prices are diffusions.

D. Kramkov and W. Schachermayer [Ann. Appl. Probab. 9 (1999) 904-950] proved the existence of log-optimal portfolios under weak assumptions in a very general setting. For many--but not all--cases, T. Goll and J. Kallsen [Stochastic Process. Appl. 89 (2000) 31-48] obtained the optimal solution explicitly in terms of the semimartingale characteristics of the price process. By extending this result, this paper provides a complete explicit characterization of log-optimal portfolios without constraints.

Moreover, the results of Goll and Kallsen are generalized here in two further respects: First, we allow for random convex trading constraints. Second, the remaining consumption time--or more generally the consumption clock--may be random, which corresponds to a life-insurance problem.

Finally, we consider neutral derivative pricing in incomplete markets.

We study the behavior of the critical price of an American put option near maturity when the underlying stock pays dividends at a continuous rate. The results also apply to foreign currencies American options.