In this paper we study a stiff problem in two-dimensional space and especially characterize its probabilistic counterpart. More specifically, consider the heat equation with a parameter $\mathit{\epsilon }>0$:

$${\partial }_{\mathit{t}}{\mathit{u}}^{\mathit{\epsilon }}(\mathit{t},\mathit{x})=\frac{1}{2}\nabla ·({\mathbf{A}}_{\mathit{\epsilon }}(\mathit{x})\nabla {\mathit{u}}^{\mathit{\epsilon }}(\mathit{t},\mathit{x})),\mathit{t}\ge 0,\mathit{x}\in {\mathbb{R}}^2,$$

where ${\mathbf{A}}_{\mathit{\epsilon }}(\mathit{x}):={\text{Id}}_2$, the identity matrix, for $\mathit{x}\notin {\mathrm{\Omega }}_{\mathit{\epsilon }}:=\{\mathit{x}=({\mathit{x}}_1,{\mathit{x}}_2)\in {\mathbb{R}}^2:|{\mathit{x}}_2|<\mathit{\epsilon }\}$ and

$${\mathbf{A}}_{\mathit{\epsilon }}(\mathit{x}):=\left(\begin{array}{cc}{\mathit{a}}_{\mathit{\epsilon }}^{-}& 0\\ 0& {\mathit{a}}_{\mathit{\epsilon }}^{|}\end{array}\right),\mathit{x}\in {\mathrm{\Omega }}_{\mathit{\epsilon }}$$

with two constants ${\mathit{a}}_{\mathit{\epsilon }}^{-},{\mathit{a}}_{\mathit{\epsilon }}^{|}>0$. The solution ${\mathit{u}}^{\mathit{\epsilon }}$ is usually called a flux. Then the stiff problem is concerned with the existence and characterization of the limit u, called the limiting flux, of ${\mathit{u}}^{\mathit{\epsilon }}$ as $\mathit{\epsilon }\downarrow 0$ in a certain sense. Note that there exists a diffusion process ${\mathit{X}}^{\mathit{\epsilon }}$ on ${\mathbb{R}}^2$ associated to this heat equation in the sense that ${\mathit{u}}^{\mathit{\epsilon }}(\mathit{t},\mathit{x}):={\mathbf{E}}^{\mathit{x}}{\mathit{u}}^{\mathit{\epsilon }}(0,{\mathit{X}}_{\mathit{t}}^{\mathit{\epsilon }})$ is its unique solution. The main result of this paper figures out the limiting process of ${\mathit{X}}^{\mathit{\epsilon }}$ as $\mathit{\epsilon }\downarrow 0$ for all possible cases. As a byproduct, the limiting flux u in an ${\mathit{L}}^2$-sense and several boundary conditions on the ${\mathit{x}}_1$-axis satisfied by u regarding various cases will be further obtained.

This paper analyzes the convergence of the finite population optimal stopping problem towards the corresponding mean field limit. Building on the viscosity solution characterization of the mean field optimal stopping problem of our previous papers (SIAM J. Control Optim. 61 (2023) 1712–1736, 2140–2164), we prove the convergence of the value functions by adapting the Barles–Souganidis (Asymptot. Anal. 4 (1991) 271–283) monotone scheme method to our context. We next characterize the optimal stopping policies of the mean field problem by the accumulation points of the finite population optimal stopping strategies. In particular, if the limiting problem has a unique optimal stopping policy, then the finite population optimal stopping strategies do converge towards this solution. As a by-product of our analysis, we provide an extension of the standard propagation of chaos to the context of stopped McKean–Vlasov diffusions.

We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form $\mathit{d}{\mathit{u}}_{\mathit{t}}-{\mathit{L}}_{\mathit{t}}({\mathit{u}}_{\mathit{t}}){\mathit{u}}_{\mathit{t}}\mathit{d}\mathit{t}={\mathit{N}}_{\mathit{t}}({\mathit{u}}_{\mathit{t}})\mathit{d}\mathit{t}\mathbf{+}\mathit{F}({\mathit{u}}_{\mathit{t}})·\mathit{d}{\mathbf{X}}_{\mathit{t}}$, where X is a γ-Hölder rough path for $\mathit{\gamma }\in (1/3,1/2)$. We explore the mild formulation that combines functional analysis techniques and controlled rough paths theory which entail the local well-posedness of such equations. We apply our results to the stochastic Landau–Lifshitz–Gilbert and Shigesada–Kawasaki–Teramoto equation.

The theory of two projections is utilized to study two-component Gibbs samplers. Through this theory, previously intractable problems regarding the asymptotic variances of two-component Gibbs samplers are reduced to elementary matrix algebra exercises. It is found that in terms of asymptotic variance, the two-component random-scan Gibbs sampler is never much worse, and could be considerably better than its deterministic-scan counterpart, provided that the selection probability is appropriately chosen. This is especially the case when there is a large discrepancy in computation cost between the two components. The result contrasts with the known fact that the deterministic-scan version has a faster convergence rate, which can also be derived from the method herein. On the other hand, a modified version of the deterministic-scan sampler that accounts for computation cost can outperform the random-scan version.

In this paper we develop tests for detecting systematic jumps in asset prices of general form, including ones not explained by observable factors, and we further propose nonparametric estimates for them. The inference is based on a panel of high-frequency asset returns, with both the sampling frequency and the size of the cross-section increasing asymptotically. The feasible limit theory developed in the paper utilizes the different asymptotic roles played by diffusive versus jump risks and systematic versus idiosyncratic risks in statistics that involve cross-sectional averages of suitably chosen transforms of the high-frequency price increments. The rate of convergence of the statistics is determined by the two asymptotically increasing dimensions of the panel, without imposing restrictions on their relative size. In an empirical application, using the developed tools, we document the existence of systematic jump risk, that is not spanned by standard (observable) risk factors, and we further show that this risk commands a nontrivial risk premium.

We consider a random walk in a truncated cone ${\mathit{K}}_{\mathit{N}}$, which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure.

The obtained results are applied to a multidimensional gambler’s problem studied by Diaconis and Ethier (Staist. Sci. 37 (2022) 289–305). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.

We investigate final outcome properties of an SIR (susceptible → infective → recovered) epidemic model defined on a population of large sub-communities in which there is stronger disease transmission within the communities than between them. Our analysis involves approximation of the epidemic process by a chain of within-community large outbreaks spreading between the communities. We derive law of large numbers and central limit type results for the number of individuals and the number of communities affected and the so-called severity of the outbreak. These results are valid as the size of communities tends to infinity, with the number of communities either fixed or also tending to infinity. The weaker between-community connections lead to randomness even in the law of large numbers type limit. As part of our proofs we also obtain a new result concerning the rate of convergence of the expected fraction infected in a standard SIR epidemic to its large-population limit.

The modified method of successive approximations (MSA) is an iterative scheme for approximating solutions to stochastic control problems in continuous time based on Pontryagin optimality principle which, starting with an initial open loop control, solves the forward equation, the backward adjoint equation and then performs a static minimization step. We observe that this is an implicit Euler scheme for a gradient flow system. We prove that appropriate interpolations of the iterates of the modified MSA converge to a gradient flow with rate τ. We then study the convergence of this gradient flow as time goes to infinity. In the general (nonconvex) case we prove that the gradient term itself converges to zero. This is a consequence of an energy identity which shows that the optimization objective decreases along the gradient flow. Moreover, in the convex case, when Pontryagin optimality principle provides a sufficient condition for optimality, we prove that the optimization objective converges at rate $\frac{1}{\mathit{S}}$ to its optimal value and at exponential rate under strong convexity. The main technical difficulties lie in obtaining appropriate properties of the Hamiltonian (growth, continuity). These are obtained by utilising the theory of bounded mean oscillation (BMO) martingales required for estimates on the adjoint backward stochastic differential equation (BSDE).

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean–Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension. As for the one-phase problems, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not necessarily monotone, so that the large system limit is obtained by a novel argument.

We introduce a simple but powerful strategy to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to nonconservative zero range processes on finite graphs, and to multi-particle random walks with positive and negative reinforcement on the edges. The results hold for a broad class of reinforcement functions, including those with superlinear growth. Our strategy consists in a comparison of the original processes with suitable reference models. To implement the comparison we estimate an object reminiscent to the Radon–Nikodym derivative on a carefully chosen set of trajectories. Our results describe the almost sure long time behaviour of the processes. We also prove a phase transition depending on the strength of the reinforcement functions.

Given a transition matrix P indexed by a finite set V of vertices, the voter model is a discrete-time Markov chain in ${\{0,1\}}^{\mathit{V}}$ where at each time-step a randomly chosen vertex x imitates the opinion of vertex y with probability $\mathit{P}(\mathit{x},\mathit{y})$. The noisy voter model is a variation of the voter model in which vertices may change their opinions by the action of an external noise. The strength of this noise is measured by an extra parameter $\mathit{p}\in [0,1]$.

In this work we analyse the density process, defined as the stationary mass of vertices with opinion 1, that is, ${\mathit{S}}_{\mathit{t}}={\sum }_{\mathit{x}\in \mathit{V}}\mathit{\pi }(\mathit{x}){\mathit{\xi }}_{\mathit{t}}(\mathit{x})$, where π is the stationary distribution of P, and ${\mathit{\xi }}_{\mathit{t}}(\mathit{x})$ is the opinion of vertex x at time t. We investigate the asymptotic behaviour of ${\mathit{S}}_{\mathit{t}}$ when t tends to infinity for different values of the noise parameter p. In particular, by allowing P and p to be functions of the size $|\mathit{V}|$, we show that, under appropriate conditions and small enough p a normalised version of ${\mathit{S}}_{\mathit{t}}$ converges to a Gaussian random variable, while for large enough p, ${\mathit{S}}_{\mathit{t}}$ converges to a Bernoulli random variable. We provide further analysis of the noisy voter model on a variety of specific graphs including the complete graph, cycle, torus, and hypercube, where we identify the critical rate p (depending on the size $|\mathit{V}|$) that separates these two asymptotic behaviours.

This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables ${\mathit{L}}_{\mathit{p}}$-estimates, $\mathit{p}\in (2,\infty )$, for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from an adjustment for the Riemann approximation based on tracking jumps of the underlying semimartingale. We discuss a way to optimize the approximation and also illustrate the sharpness of the obtained convergence rates. When the small jump activity of the semimartingale behaves like an α-stable process with $\mathit{\alpha }\in (1,2)$, our scheme achieves under a regular regime the same convergence rate for the error as in Rosenbaum and Tankov [Ann. Appl. Probab. 24 (2014) 1002–1048]. Moreover, our approach extends to the case $\mathit{\alpha }\in (0,1]$ and to the ${\mathit{L}}_{\mathit{p}}$-setting which are not treated there. As an application, we apply the methods in the special case where the semimartingale is an exponential Lévy process to mean-variance hedging of European type options.

Multitudinous probabilistic and combinatorial objects are associated with generating functions satisfying a composition scheme $\mathit{F}(\mathit{z})=\mathit{G}(\mathit{H}(\mathit{z}))$. The analysis becomes challenging when this scheme is critical (i.e., G and H are simultaneously singular). Motivated by many examples (random mappings, planar maps, directed lattice paths), we consider a natural extension of this scheme, namely $\mathit{F}(\mathit{z},\mathit{u})=\mathit{G}(\mathit{u}\mathit{H}(\mathit{z}))\mathit{M}(\mathit{z})$. We also consider a variant of this scheme, which allows us to analyse the number of H-components of a given size in F.

We prove that these two models lead to a rich world of limit laws, where we identify the key role played by a new universal law introduced in this article: the three-parameter Mittag-Leffler distribution, which is essentially the product of a beta and a Mittag-Leffler distribution. We also prove (double) phase transitions, additionally involving Boltzmann and mixed Poisson distributions, bringing a unified explanation of the associated thresholds. In all cases we obtain moment convergence and local limit theorems. We end with extensions of the critical composition scheme to a cycle scheme and to the multivariate case, leading to product distributions. Applications are presented for random walks, trees (supertrees of trees, increasingly labelled trees, preferential attachment trees), triangular Pólya urns, and the Chinese restaurant process.

A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.

This paper introduces a notion of viscosity solutions for second order elliptic Hamilton–Jacobi–Bellman (HJB) equations with infinite delay associated with infinite-horizon optimal control problems for stochastic differential equations with infinite delay. We identify the value functional of optimal control problems as unique viscosity solution to associated second order elliptic HJB equations with infinite delay. We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property.

In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for which the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system possesses nonlinearity of type $\mathit{f}(|\mathit{y}|)|\mathit{z}{|}^2$, where f is any locally integrable function. We prove both the Malliavin and classical differentiability of solutions to this type of QFBSDEs and provide representations of these derivatives processes. As a by-product, we derive a representation formula of the control variable ${\mathit{Z}}_{\mathit{t}}$ as a conditional expectation of the terminal value, the driver and the Malliavin weights, when the drift term is only bounded and Hölder continuous. We study a numerical approximation of this system in the sense of Imkeller and Dos Reis (Stochastic Process. Appl. 120 (2010) 2286–2288) in which the authors assume that the drift is Lipschitz and the driver of the BSDE is globally Lipschitz in y and quadratic in the traditional sense in z (i.e., f is a positive constant). We show that the rate of convergence is the same as in (Stochastic Process. Appl. 120 (2010) 2286–2288).

We consider empirical measures of ${\mathbb{R}}^{\mathit{d}}$-valued stochastic process in finite discrete-time. We show that the adapted empirical measure introduced in the recent work (Ann. Appl. Probab. 32 (2022) 529–550) by Backhoff et al. in compact spaces can be defined analogously on ${\mathbb{R}}^{\mathit{d}}$, and that it converges almost surely to the underlying measure under the adapted Wasserstein distance. Moreover, we quantitatively analyze the convergence of the adapted Wasserstein distance between those two measures. We establish convergence rates of the expected error as well as the deviation error under different moment conditions. Under suitable integrability and kernel assumptions, we recover the optimal convergence rates of both expected error and deviation error. Furthermore, we propose a modification of the adapted empirical measure with projection on a nonuniform grid, which obtains the same convergence rate but under weaker assumptions.

We propose a new Monte Carlo-based estimator for digital options with assets modelled by a stochastic differential equation (SDE). The new estimator is based on repeated path splitting and relies on the correlation of approximate paths of the underlying SDE that share parts of a Brownian path. Combining this new estimator with multilevel Monte Carlo (MLMC) leads to an estimator with a computational complexity that is similar to the complexity of a MLMC estimator when applied to options with Lipschitz payoffs.

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we investigate the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then, we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs.

We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle systems are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous’ spectral gap conjecture originally formulated for the interchange process. Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process.

Consider a multiple hypothesis testing setting involving rare/weak effects: relatively few tests, out of possibly many, deviate from their null hypothesis behavior. Summarizing the significance of each test by a p-value, we construct a global test against the joint null using the higher criticism (HC) statistics of these p-values. We calibrate the rare/weak model using parameters controlling the asymptotic distribution of nonnull p-values near zero. We derive a region in the parameter space where the HC test is asymptotically powerless. Our derivation involves very different tools than previously used to show powerlessness of HC, relying on properties of the empirical processes underlying HC. In particular, our result applies to situations where HC is not asymptotically optimal, or when the asymptotically detectable region of the parameter space is unknown.

We consider the so-called directed configuration model (DCM), that is, a random directed graph with prescribed in- and out-degrees. In this random geometry, we study the meeting time of two random walks starting at stationarity, the coalescence time for a system of coalescent random walks, and the consensus time of the classical voter model. Indeed, it is known that the latter three quantities are related to each other under certain mean field conditions requiring fast enough mixing time and not too concentrated stationary distribution. Our main result shows that, for a typical large graph from the DCM ensemble, the meeting time is well-approximated by an exponential random variable for which we provide the first-order asymptotics of its expectation, showing that the latter is linear in the size of the graph, and its pre-constant depends on some explicit statistics of the degree sequence. As a byproduct, we explore the effect of the degree sequence in changing the meeting, coalescence, and consensus time by discussing several classes of examples of interest also from an applied perspective. Our approach follows the classical idea of converting meeting into hitting times of a proper collapsed chain, which we control by the so-called first visit time lemma. The main technical challenge is related to the fact that in such a directed setting the stationary distribution is random, and it depends on the whole realization of the graph. As a consequence, a good share of the proof focuses on showing that certain functions of the stationary distribution concentrate around their expectations, and on their characterization, via proper annealing arguments.