Let $Y_{i}$, $i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset\mathbb{R}^{d}$ and consider sums $\sum_{i=1}^{n}\xi(n^{1/m}Y_{i},\{n^{1/m}Y_{j}\}_{j=1}^{n})$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal{Y} )$, with $y\in\mathbb{R}^{d}$ and $\mathcal{Y}\subset\mathbb{R}^{d}$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal{M} $. We apply the general results to establish the limit theory of dimension and volume content estimators, Rényi and Shannon entropy estimators and clique counts in the Vietoris–Rips complex on $\{Y_{i}\}_{i=1}^{n}$.

We study the problem of super-replication for game options under proportional transaction costs. We consider a multidimensional continuous time model, in which the discounted stock price process satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy. This result is an extension of previous papers (see [Statist. Decisions 27 (2009) 357–369] and [Ann. Appl. Probab. 18 (2008) 491–520]) which considered only European options. In these papers the authors showed that with the presence of proportional transaction costs the super-replication price of a European option is given in terms of the concave envelope of the payoff function. In the present work we prove that for game options the super-replication price is given by a game variant analog of the standard concave envelope term. The treatment of game options is more complicated and requires additional tools. We combine the theory of consistent price systems together with the theory of extended weak convergence which was developed in [Weak convergence of stochastic processes for processes viewed in the strasbourg manner (1981) Preprint]. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox–Ingersoll–Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox–Ingersoll–Ross processes and Bessel processes.

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\varepsilon}$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.

This paper concerns optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$. More precisely, we are interested in modifications of the Shepp–Shiryaev optimal stopping problem [Avram, Kyprianou and Pistorius Ann. Appl. Probab. 14 (2004) 215–238; Shepp and Shiryaev Ann. Appl. Probab. 3 (1993) 631–640; Shepp and Shiryaev Theory Probab. Appl. 39 (1993) 103–119]. First, we consider a capped version of the Shepp–Shiryaev optimal stopping problem and provide the solution explicitly in terms of scale functions. In particular, the optimal stopping boundary is characterised by an ordinary differential equation involving scale functions and changes according to the path variation of $X$. Secondly, in the spirit of [Shepp, Shiryaev and Sulem Advances in Finance and Stochastics (2002) 271–284 Springer], we consider a modification of the capped version of the Shepp–Shiryaev optimal stopping problem in the sense that the decision to stop has to be made before the process $X$ falls below a given level.

We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump–Mode–Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes.

We consider the FCFS $GI/G/n$ queue in the so-called Halfin–Whitt heavy traffic regime. We prove that under minor technical conditions the associated sequence of steady-state queue length distributions, normalized by $n^{1/2}$, is tight. We derive an upper bound on the large deviation exponent of the limiting steady-state queue length matching that conjectured by Gamarnik and Momcilovic [Adv. in Appl. Probab. 40 (2008) 548–577]. We also prove a matching lower bound when the arrival process is Poisson.

Our main proof technique is the derivation of new and simple bounds for the FCFS $GI/G/n$ queue. Our bounds are of a structural nature, hold for all $n$ and all times $t\geq0$, and have intuitive closed-form representations as the suprema of certain natural processes which converge weakly to Gaussian processes. We further illustrate the utility of this methodology by deriving the first nontrivial bounds for the weak limit process studied in [Ann. Appl. Probab. 19 (2009) 2211–2269].

In this article, we build upon the work of Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149–190] to define a notion of a second order backward stochastic differential equation reflected on a lower càdlàg obstacle. We prove existence and uniqueness of the solution under a Lipschitz-type assumption on the generator, and we investigate some links between our reflected 2BSDEs and nonclassical optimal stopping problems. Finally, we show that reflected 2BSDEs provide a super-hedging price for American options in a market with volatility uncertainty.

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson–Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.

Under multiplicative drift and other regularity conditions, it is established that the asymptotic variance associated with a particle filter approximation of the prediction filter is bounded uniformly in time, and the nonasymptotic, relative variance associated with a particle approximation of the normalizing constant is bounded linearly in time. The conditions are demonstrated to hold for some hidden Markov models on noncompact state spaces. The particle stability results are obtained by proving $v$-norm multiplicative stability and exponential moment results for the underlying Feynman–Kac formulas.

The Axelrod model is a spatial stochastic model for the dynamics of cultures which includes two important social factors: social influence, the tendency of individuals to become more similar when they interact, and homophily, the tendency of individuals to interact more frequently with individuals who are more similar. Each vertex of the interaction network is characterized by its culture, a vector of $F$ cultural features that can each assumes $q$ different states. Pairs of neighbors interact at a rate proportional to the number of cultural features they have in common, which results in the interacting pair having one more cultural feature in common. In this article, we continue the analysis of the Axelrod model initiated by the first author by proving that the one-dimensional system fixates when $F\leq cq$ where the slope satisfies the equation $e^{-c}=c$. In addition, we show that the two-feature model with at least three states fixates. This last result is sharp since it is known from previous works that the one-dimensional two-feature two-state Axelrod model clusters.

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_{n}(\xi)$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf{E} [{C_{n}(\xi)}]\sim\kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_{n}(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_{n}(x)$. Our results permit us to describe a limit process for the costs $C_{n}(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf{E} [{\max_{x\in[0,1]}C_{n}(x)}]\sim\gamma n^{\beta}$; this settles a question of Devroye [Pers. Comm., 2000].