Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.

We construct a time-consistent sublinear expectation in the setting of volatility uncertainty. This mapping extends Peng’s $G$-expectation by allowing the range of the volatility uncertainty to be stochastic. Our construction is purely probabilistic and based on an optimal control formulation with path-dependent control sets.

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution $\pi$ on a given finite partially ordered state space ${ \mathcal{X}}$. When $K\preceq L$ in this partial order we say that $K$ and $L$ satisfy a comparison inequality. We establish that if $K_{1},\ldots,K_{t}$ and $L_{1},\ldots,L_{t}$ are reversible and $K_{s}\preceq L_{s}$ for $s=1,\ldots,t$, then $K_{1}\cdots K_{t}\preceq L_{1}\cdots L_{t}$. In particular, in the time-homogeneous case we have $K^{t}\preceq L^{t}$ for every $t$ if $K$ and $L$ are reversible and $K\preceq L$, and using this we show that (for suitable common initial distributions) the Markov chain $Y$ with kernel $K$ mixes faster than the chain $Z$ with kernel $L$, in the strong sense that at every time $t$ the discrepancy—measured by total variation distance or separation or $L^{2}$-distance—between the law of $Y_{t}$ and $\pi$ is smaller than that between the law of $Z_{t}$ and $\pi$.

Using comparison inequalities together with specialized arguments to remove the stochastic monotonicity restriction, we answer a question of Persi Diaconis by showing that, among all symmetric birth-and-death kernels on the path ${ \mathcal{X}}=\{0,\ldots,n\}$, the one (we call it the uniform chain) that produces fastest convergence from initial state $0$ to the uniform distribution has transition probability $1/2$ in each direction along each edge of the path, with holding probability $1/2$ at each endpoint.

We also use comparison inequalities:

(i) to identify, when $\pi$ is a given log-concave distribution on the path, the fastest-mixing stochastically monotone birth-and-death chain started at $0$, and

(ii) to recover and extend a result of Peres and Winkler that extra updates do not delay mixing for monotone spin systems.

Among the fastest-mixing chains in (i), we show that the chain for uniform $\pi$ is slowest in the sense of maximizing separation at every time.

This paper resolves a question proposed in Kardaras and Robertson [Ann. Appl. Probab. 22 (2012) 1576–1610]: how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of nondominated probability measures.

In join the shortest queue networks, incoming jobs are assigned to the shortest queue from among a randomly chosen subset of $D$ queues, in a system of $N$ queues; after completion of service at its queue, a job leaves the network. We also assume that jobs arrive into the system according to a rate-$\alpha N$ Poisson process, $\alpha<1$, with rate-$1$ service at each queue. When the service at queues is exponentially distributed, it was shown in Vvedenskaya et al. [Probl. Inf. Transm. 32 (1996) 15–29] that the tail of the equilibrium queue size decays doubly exponentially in the limit as $N\rightarrow\infty$. This is a substantial improvement over the case $D=1$, where the queue size decays exponentially.

The reasoning in [Probl. Inf. Transm. 32 (1996) 15–29] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating general service time distributions was introduced in Bramson et al. [In Proc. ACM SIGMETRICS (2010) 275–286]. The program relies on an ansatz that asserts, in equilibrium, any fixed number of queues become independent of one another as $N\rightarrow\infty$. This ansatz was demonstrated in several settings in Bramson et al. [Queueing Syst. 71 (2012) 247–292], including for networks where the service discipline is FIFO and the service time distribution has a decreasing hazard rate.

In this article, we investigate the limiting behavior, as $N\rightarrow\infty$, of the equilibrium at a queue when the service discipline is FIFO and the service time distribution has a power law with a given exponent $-\beta$, for $\beta>1$. We show under the above ansatz that, as $N\rightarrow\infty$, the tail of the equilibrium queue size exhibits a wide range of behavior depending on the relationship between $\beta$ and $D$. In particular, if $\beta>D/(D-1)$, the tail is doubly exponential and, if $\beta<D/(D-1)$, the tail has a power law. When $\beta=D/(D-1)$, the tail is exponentially distributed.

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the $M/M/n$ queue in the Halfin–Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant $B^{\ast}\approx1.85772\mbox{ s.t.}$ when a certain excess parameter $B\in(0,B^{\ast}]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^{2}}{4}$. For $B>B^{\ast}$, the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed $n$ by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer].

We also prove explicit bounds on the distance to stationarity for the $M/M/n$ queue in the Halfin–Whitt regime, when $B<B^{\ast}$. Our bounds scale independently of $n$ in the Halfin–Whitt regime, and do not follow from the weak-convergence theory.

The Euler–Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler’s method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does—in contrast to classical Monte Carlo methods—not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case.

We analyze kinetically constrained $0$–$1$ spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson–Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice ${\mathbb{Z}}^{d}$.

In this paper we consider a method of solving optimal stopping problems in discrete and continuous time based on their dual representation. A novel and generic simulation-based optimization algorithm not involving nested simulations is proposed and studied. The algorithm involves the optimization of a genuinely penalized dual objective functional over a class of adapted martingales. We prove the convergence of the proposed algorithm and demonstrate its efficiency for optimal stopping problems arising in option pricing.

The Azéma–Yor solution (resp., the Perkins solution) of the Skorokhod embedding problem has the property that it maximizes (resp., minimizes) the law of the maximum of the stopped process. We show that these constructions have a wider property in that they also maximize (and minimize) expected values for a more general class of bivariate functions $F(W_{\tau},S_{\tau})$ depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions $g$, they also maximize and minimize $\mathbb{E} [\int_{0}^{\tau}g(S_{t})\,dt]$ amongst embeddings of $\mu$, although, perhaps surprisingly, we show that for increasing $g$ the Azéma–Yor embedding minimizes this quantity, and the Perkins embedding maximizes it.

For $g(s)=s^{-2}$ we show how these results are useful in calculating model independent bounds on the prices of variance swaps.

Along the way we also consider whether $\mu_{n}$ converges weakly to $\mu$ is a sufficient condition for the associated Azéma–Yor and Perkins stopping times to converge. In the case of the Azéma–Yor embedding, if the potentials at zero also converge, then the stopping times converge almost surely, but for the Perkins embedding this need not be the case. However, under a further condition on the convergence of atoms at zero, the Perkins stopping times converge in probability (and hence converge almost surely down a subsequence).

Near critical catalyst-reactant branching processes with controlled immigration are studied. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous time branching process; in addition there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. Such models are motivated by problems in chemical kinetics where one wants to keep the level of a catalyst above a certain threshold in order to maintain a desired level of reaction activity. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst. Stochastic averaging principles under fast catalyst dynamics are established. In the case where the catalyst evolves “much faster” than the reactant, a scaling limit, in which the reactant is described through a one dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained. Proofs rely on constrained martingale problem characterizations, Lyapunov function constructions, moment estimates that are uniform in time and the scaling parameter and occupation measure techniques.

We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools, mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a natural (load balancing) routing/scheduling rule, Longest-Queue Freest-Server (LQFS-LB), in the many-server asymptotic regime, such that the exogenous arrival rates of the customer classes, as well as the number of agents in each pool, grow to infinity in proportion to some scaling parameter $r$. Equilibrium point of the system under LQBS-LB is the desired operating point, with server pool loads minimized and perfectly balanced.

Our main results are as follows. (a) We show that, quite surprisingly (given the tree assumption), for certain parameter ranges, the fluid limit of the system may be unstable in the vicinity of the equilibrium point; such instability may occur if the activity graph is not “too small.” (b) Using (a), we demonstrate that the sequence of stationary distributions of diffusion-scaled processes [measuring $O(\sqrt{r})$ deviations from the equilibrium point] may be nontight, and in fact may escape to infinity. (c) In one special case of interest, however, we show that the sequence of stationary distributions of diffusion-scaled processes is tight, and the limit of stationary distributions is the stationary distribution of the limiting diffusion process.

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_{t}$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_{s},s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $\theta^{f}_{t}$, defined on the space of continuous paths $C([0,t],\mathbb{R} ^{d})$ endowed with the uniform convergence topology such that $\pi_{t}(f)=\theta^{f}_{t}(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43–56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125–139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160–167], Kushner [Stochastics 3 (1979) 75–83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260–278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505–528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403–424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is “lifted” to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding Lévy area process, and we show that there exists a continuous map $\theta_{t}^{f}$, defined on a suitably chosen space of Hölder continuous paths such that $\pi_{t}(f)=\theta_{t}^{f}(\mathbf{Y})$, almost surely.