This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic $\mathsf{L}^{p}$ error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM.

We give a probabilistic interpretation of the Monte Carlo scheme proposed by Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322–1364] for fully nonlinear parabolic PDEs, and hence generalize it to the path-dependent (or non-Markovian) case for a general stochastic control problem. A general convergence result is obtained by a weak convergence method in the spirit of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (1992) Springer]. We also get a rate of convergence using the invariance principle technique as in Dolinsky [Electron. J. Probab. 17 (2012) 1–5], which is better than that obtained by viscosity solution method. Finally, by approximating the conditional expectations arising in the numerical scheme with simulation-regression method, we obtain an implementable scheme.

The card-cyclic-to-random shuffle on $n$ cards is defined as follows: at time $t$ remove the card with label $t$ mod $n$ and randomly reinsert it back into the deck. Pinsky [Probabilistic and combinatorial aspects of the card-cyclic-to-random shuffle (2011). Unpublished manuscript] introduced this shuffle and asked how many steps are needed to mix the deck. He showed $n$ steps do not suffice. Here we show that the mixing time is on the order of $\Theta(n\log n)$.

We derive limit theorems for the empirical distribution function of “devolatilized” increments of an Itô semimartingale observed at high frequencies. These “devolatilized” increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and “large” jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Itô semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Itô semimartingales with nonvanishing diffusion coefficient against Itô semimartingales with no diffusion component.

We consider the Williams–Bjerknes model, also known as the biased voter model on the $d$-regular tree $\mathbb{T} ^{d}$, where $d\geq3$. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate $\lambda\geq1$, while healthy vertices heal their neighbors at Poisson rate $1$. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability if and only if $\lambda>1$. We show that there exists a threshold $\lambda_{c}\in(1,\infty)$ such that if $\lambda>\lambda_{c}$ then in the above setting with positive probability, all vertices will become eventually infected forever, while if $\lambda<\lambda_{c}$, all vertices will become eventually healthy with probability $1$. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on $\mathbb{T} ^{d}$—above $\lambda_{c}$. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of $\mathbb{T} ^{d}$.

This paper studies diffusion processes constrained to the positive orthant under infinitesimal changes in the drift. Our first main result states that any constrained function and its (left) drift-derivative is the unique solution to an augmented Skorohod problem. Our second main result uses this characterization to establish a basic adjoint relationship for the stationary distribution of the constrained diffusion process jointly with its left-derivative process.

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf{R} ^{m}$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf{Z} ^{d}$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<\alpha<d$), we address the convergence as $N\to\infty$ of the empirical measure of the diffusions to the solution of a deterministic McKean–Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when $\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.

In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein, Schoenmakers and Spokoiny [Bernoulli 10 (2004) 281–312] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-$N$ accuracy, and hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset.

A multi-class single-server system with general service time distributions is studied in a moderate deviation heavy traffic regime. In the scaling limit, an optimal control problem associated with the model is shown to be governed by a differential game that can be explicitly solved. While the characterization of the limit by a differential game is akin to results at the large deviation scale, the analysis of the problem is closely related to the much studied area of control in heavy traffic at the diffusion scale.

Consider a $d$-ary rooted tree ($d\geq3$) where each edge $e$ is assigned an i.i.d. (bounded) random variable $X(e)$ of negative mean. Assign to each vertex $v$ the sum $S(v)$ of $X(e)$ over all edges connecting $v$ to the root, and assume that the maximum $S_{n}^{*}$ of $S(v)$ over all vertices $v$ at distance $n$ from the root tends to infinity (necessarily, linearly) as $n$ tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature $1/\beta$ of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous [Algorithmica 22 (1998) 388–412]. The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree.

We study an admissions control problem, where a queue with service rate $1-p$ receives incoming jobs at rate $\lambda\in(1-p,1)$, and the decision maker is allowed to redirect away jobs up to a rate of $p$, with the objective of minimizing the time-average queue length.

We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate $\sim\log_{1/(1-p)}\frac{1}{1-\lambda}$, as $\lambda\to1$. In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, $(1-p)/p$, as $\lambda\to1$. We further show that the finite limit of $(1-p)/p$ can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as $\mathcal{O}(\log\frac{1}{1-\lambda})$, as $\lambda\to1$. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.

In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\mathop{=}^\mathcal{D}f(V)$, where $f(v)=A\max\{v,D\}+B$ for $(A,B,D)\in(0,\infty)\times\mathbb{R}^{2}$. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide class of dynamic importance sampling estimators. Moreover, using extensions of ideas from nonlinear renewal theory, we provide a precise description of the running time of the algorithm. To establish these results, we develop new techniques concerning the convergence of moments of stopped perpetuity sequences, and the first entrance and last exit times of associated Markov chains on $\mathbb{R}$. We illustrate our methods with a variety of numerical examples which demonstrate the ease and scope of the implementation.

A general method to construct recombinant tree approximations for stochastic volatility models is developed and applied to the Heston model for stock price dynamics. In this application, the resulting approximation is a four tuple Markov process. The first two components are related to the stock and volatility processes and take values in a two-dimensional binomial tree. The other two components of the Markov process are the increments of random walks with simple values in $\{-1,+1\}$. The resulting efficient option pricing equations are numerically implemented for general American and European options including the standard put and calls, barrier, lookback and Asian-type pay-offs. The weak and extended weak convergences are also proved.