The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\zeta$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_{s}>b(s),0\leq s\leq t\}=\mathbb{P}\{\zeta>t\}$. We study a “smoothed” version of this problem and ask whether there is a “barrier” $b$ such that $\mathbb{E}[\exp(-\lambda\int_{0}^{t}\psi(B_{s}-b(s))\,ds)]=\mathbb{P}\{\zeta>t\}$, where $\lambda$ is a killing rate parameter, and $\psi:\mathbb{R}\to[0,1]$ is a nonincreasing function. We prove that if $\psi$ is suitably smooth, the function $t\mapsto\mathbb{P}\{\zeta>t\}$ is twice continuously differentiable, and the condition $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ holds for the hazard rate of $\zeta$, then there exists a unique continuously differentiable function $b$ solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.

The Wang–Landau algorithm aims at sampling from a probability distribution, while penalizing some regions of the state space and favoring others. It is widely used, but its convergence properties are still unknown. We show that for some variations of the algorithm, the Wang–Landau algorithm reaches the so-called flat histogram criterion in finite time, and that this criterion can be never reached for other variations. The arguments are shown in a simple context—compact spaces, density functions bounded from both sides—for the sake of clarity, and could be extended to more general contexts.

This paper consists of two parts. In the first part we prove the fundamental theorem of asset pricing under short sales prohibitions in continuous-time financial models where asset prices are driven by nonnegative, locally bounded semimartingales. A key step in this proof is an extension of a well-known result of Ansel and Stricker. In the second part we study the hedging problem in these models and connect it to a properly defined property of “maximality” of contingent claims.

We consider a flow-level model of a network operating under an $\alpha$-fair bandwidth sharing policy (with $\alpha>0$) proposed by Roberts and Massoulié [Telecomunication Systems 15 (2000) 185–201]. This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network.

We study the transient properties as well as the steady-state distribution of the model. In particular, for $\alpha\geq1$, we obtain bounds on the maximum number of flows in the network over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property for all $\alpha\geq1$.

For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $\alpha>0$, by relying on a norm-like Lyapunov function. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009) 1719–1780], in steady state, for the case where $\alpha=1$ and under a local traffic condition.

We determine the mixing time of a simple Gibbs sampler on the unit simplex, confirming a conjecture of Aldous. The upper bound is based on a two-step coupling, where the first step is a simple contraction argument and the second step is a non-Markovian coupling. We also present a MCMC-based perfect sampling algorithm based on our proof which can be applied with Gibbs samplers that are harder to analyze.

We derive new results comparing the asymptotic variance of diffusions by writing them as appropriate limits of discrete-time birth–death chains which themselves satisfy Peskun orderings. We then apply our results to simulated tempering algorithms to establish which choice of inverse temperatures minimises the asymptotic variance of all functionals and thus leads to the most efficient MCMC algorithm.

We prove a complete convergence theorem for a class of symmetric voter model perturbations with annihilating duals. A special case of interest covered by our results is the stochastic spatial Lotka–Volterra model introduced by Neuhauser and Pacala [Ann. Appl. Probab. 9 (1999) 1226–1259]. We also treat two additional models, the “affine” and “geometric” voter models.

We consider directed last-passage percolation on the random graph $G=(V,E)$ where $V=\mathbb{Z}$ and each edge $(i,j)$, for $i<j\in\mathbb{Z}$, is present in $E$ independently with some probability $p\in (0,1]$. To every $(i,j)\in E$ we attach i.i.d. random weights $v_{i,j}>0$. We are interested in the behaviour of $w_{0,n}$, which is the maximum weight of all directed paths from $0$ to $n$, as $n\rightarrow\infty$. We see two very different types of behaviour, depending on whether $\mathbb{E}[v_{i,j}^{2}]<\infty$ or $\mathbb{E}[v_{i,j}^{2}]=\infty$. In the case where $\mathbb{E}[v_{i,j}^{2}]<\infty$ we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where $\mathbb{E}[v_{i,j}^{2}]=\infty$ we obtain scaling laws and asymptotic distributions expressed in terms of a “continuous last-passage percolation” model on $[0,1]$; these are related to corresponding results for two-dimensional last-passage percolation with heavy-tailed weights obtained in Hambly and Martin [Probab. Theory Related Fields 137 (2007) 227–275].

First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for the BMC and apply them to derive a second result on the MDP for a bounded functional of the BMC with a larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (for either the Gaussian setting or the bounded setting), and the MDP for least square estimators of the parameters of a first-order bifurcating autoregressive process.

Consider the random process ${(X_{t})}_{t\geq0}$ solution of $\dot{X}_{t}=A_{I_{t}}X_{t}$, where ${(I_{t})}_{t\geq0}$ is a Markov process on $\{0,1\}$, and $A_{0}$ and $A_{1}$ are real Hurwitz matrices on $\mathbb{R}^{2}$. Assuming that there exists $\lambda\in(0,1)$ such that $(1-\lambda)A_{0}+\lambda A_{1}$ has a positive eigenvalue, we establish that $\|X_{t}\|$ may converge to 0 or $+\infty$ depending on the jump rate of the process $I$. An application to product of random matrices is studied. This paper can be viewed as a probabilistic counterpart of the paper [Internat. J. Control 82 (2009) 1882–1888] by Balde, Boscain and Mason.

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induces a new proof of the optimality of Azéma–Yor solution of the SEP for a certain class of lookback options. Unlike the SEP technique, our approach applies to a large class of exotics and is suitable for numerical approximation techniques.

The Metropolis-adjusted Langevin algorithm (MALA) is a Metropolis–Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich–Rubinstein–Wasserstein distance of the MALA chain with semi-implicit Euler proposals applied to log-concave probability measures that have a density w.r.t. a Gaussian reference measure. For sufficiently “regular” densities, the estimates are dimension-independent, and they hold for sufficiently small step sizes $h$ that do not depend on the dimension either. In the limit $h\downarrow0$, the bounds approach the known optimal contraction rates for overdamped Langevin diffusions in a convex potential.

A similar approach also applies to Metropolis–Hastings chains with Ornstein–Uhlenbeck proposals. In this case, the resulting estimates are still independent of the dimension but less optimal, reflecting the fact that MALA is a higher order approximation of the diffusion limit than Metropolis–Hastings with Ornstein–Uhlenbeck proposals.

A many-server heavy-traffic FCLT is proved for the $G_{t}/M/s_{t}+\mathit{GI} $ queueing model, having time-varying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical support for the approximating deterministic fluid model the authors analyzed in a previous paper and a refined Gaussian process approximation, using variance formulas given here. The model is assumed to alternate between underloaded and overloaded intervals, with critical loading only at the isolated switching points. The proof is based on a recursive analysis of the system over these successive intervals, drawing heavily on previous results for infinite-server models. The FCLT requires careful treatment of the initial conditions for each interval.

In this paper we prove that the asymptotic rate of exponential loss of memory of a finite state hidden Markov model is bounded above by the difference of the first two Lyapunov exponents of a certain product of matrices. We also show that this bound is in fact realized, namely for almost all realizations of the observed process we can find symbols where the asymptotic exponential rate of loss of memory attains the difference of the first two Lyapunov exponents. These results are derived in particular for the observed process and for the filter; that is, for the distribution of the hidden state conditioned on the observed sequence. We also prove similar results in total variation.