We consider the filtering problem associated with partially observed McKean–Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times; the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem is challenging even in the ordinary SDE case and requires numerical approximations. Based on the ideas in Ben Rached et al. (2024) and dos Reis et al. (2023), we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the aforementioned expectations. We prove under assumptions that, for $\varepsilon>0$, to obtain a mean square error of $\mathcal{O}(\varepsilon^2)$ the PF has a cost per observation time of $\mathcal{O}(\varepsilon^{-5})$, and the MLPF costs $\mathcal{O}(\varepsilon^{-4})$ (best case) or $\mathcal{O}(\varepsilon^{-4}\log(\varepsilon)^2)$ (worst case). Our theoretical results are supported by numerical experiments.

We investigate the EM approximation for $\mathbb{R}^d$-valued ergodic stochastic differential equations (SDEs) driven by rotationally invariant $\alpha$-stable processes ($\alpha\in(1,2)$) with Markovian switching. The coefficient g violates the dissipative condition for certain states of the switching process. Using the Lindeberg principle, we establish quantitative error bounds between the original process $(X_t,R_t)_{t\geqslant 0}$ and its Euler–Maruyama (EM) scheme under a specially designed metric. Furthermore, we derive both a central limit theorem and a moderate derivation principle for the empirical measures of both the SDE and its EM scheme. The theoretical results are subsequently validated through a concrete example.

The purpose of this paper is to analyze the degree index and the clustering index in dense random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this paper. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős–Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. In addition to the Erdős–Rényi case, we also present a simulation-based analysis for random regular graphs, the Barabási–Albert model, and the Watts–Strogatz model.

We consider general discrete-time multitype branching processes on a countable set X. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in X, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalising the germ order introduced by Hutchcroft, which relies on the generating function of the process. We prove that given two multitype branching processes with laws ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ respectively, with ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$, then in every set where there is survival according to ${\boldsymbol{\nu}}$, there is also survival according to ${\boldsymbol{\mu}}$. Moreover, in every set where there is strong survival according to ${\boldsymbol{\nu}}$, there is also strong survival according to ${\boldsymbol{\mu}}$, provided that the supremum of the global extinction probabilities for the ${\boldsymbol{\nu}}$ process, taken over all starting points x, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.

Let $\mathrm{P}_K(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in K, a compact convex polygon in $\mathbb{R}^2$ with non-empty interior, are in convex position, that is, form the vertex set of a convex polygon. In this paper, the exact asymptotic behaviour of $\mathrm{P}_K(n)$ is determined as $n\to+\infty$. This improves on a famous result of Bárány [1999 Ann. Probab. 27, 2020–2034] (yet valid for a general convex set K) and a result initiated in the case where K is a regular convex polygon (Morin [2025 Adv. Appl. Probab. 57, 811–870]).

We consider d-dimensional stochastic differential equations (SDEs) of the form $\textrm{d}U_t = b(U_t)\,\textrm{d}t + \sigma\,\textrm{d}Z_t$. Let $X_t$ denote the solution if the driving noise $Z_t$ is a d-dimensional rotationally symmetric $\alpha$-stable process ($1\lt \alpha\lt 2$), and let $Y_t$ be the solution if the driving noise is a d-dimensional Brownian motion. Continuing the work started in Deng et al. (2025), we derive an estimate of the total variation distance $\|\textrm{law}(X_{t})-\textrm{law}(Y_{t})\|_\textrm{TV}$ for all $t \gt 0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t$ and $Y_t$, respectively, satisfy $\|\mu_\alpha-\mu_2\|_\textrm{TV} \leq {Cd\log(1+d)}(2-\alpha)/({\alpha-1})$. We show that this bound is optimal with respect to $\alpha$ by an Ornstein–Uhlenbeck SDE. Combining this bound with a recent interpolation result from Huang et al. (2023), we can derive a bound in the Wasserstein-p distance ($0 \lt p \lt 1$): $\|\mu_\alpha-\mu_2\|_{W_p} \leq {Cd^{(p+3)/2}\log(1+d)}(2-\alpha)/{\alpha-1}$.

We consider the asymptotic behaviour of the expectation of the perimeter deviation of a uniform random spherical disc–polygon in a spherical spindle convex disc with smooth boundary. We also introduce the notion of duality on the sphere, define a model of random circumscribed disc–polygons, and determine some asymptotic results about them.

In this paper, we study the self-normalized Cramér-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry–Esseen bound for the SGLD. Our approach is by constructing a stochastic differential equation to approximate the SGLD and then applying Stein’s method to decompose the empirical measure into a martingale difference series sum and a negligible remainder term.

This paper introduces a novel expectation-maximization (EM) algorithm for estimating general phase-type (PH) distributions from left-truncated and right-censored (LTRC) data, a common challenge in survival analysis. The proposed algorithm is highly efficient with computational complexity that scales with the number of nonzero elements in the generator matrix. This feature makes the estimation of high-dimensional, sparse PH models computationally tractable and enables the practical use of the computationally intensive extended information criterion for model selection. Numerical experiments demonstrate its significant speed advantage over a modern benchmark and the applicability of PH models to complex lifetime data.

Consider a random walk in a time-inhomogeneous random environment. When the environment is stationary and ergodic, we identify a quenched harmonic function for almost every realization of the environment. This function allows us to define a random walk in a random environment conditioned to stay positive, using Doob’s h-transform.

We study large deviations for Cox–Ingersoll–Ross processes with small noise and state-dependent fast switching via associated Hamilton–Jacobi–Bellman equations. As time scales separate, when the noise goes to 0 and the rate of switching goes to $\infty$, we get a limit equation characterized by the averaging principle. Moreover, we prove the large deviation principle with an action-integral form rate function to describe the asymptotic behavior of such systems. The new ingredient is establishing the comparison principle in the singular context. The proof is carried out using the nonlinear semigroup method from Feng and Kurtz’s book [14].

Fractional Brownian motion, with its long-time correlated increments, has been applied in many fields in recent years. Since volatility was shown to be rough by Gatheral, Jaisson, and Rosenbaum, fractional Brownian motion has gained popularity as a financial model. In this work, we revisit the definitions and properties of the univariate and multivariate fractional Brownian motions, and consider four simulation methods. We demonstrate the issues associated with applying the standard Euler scheme for simulating stochastic processes driven by fractional Brownian motion with $H < \frac{1}{2}$ (which we call the rough models). We then introduce a novel approximate method for simulating such rough models based on the fast algorithm by Ma and Wu, which accounts for a factor of 10 speedup. Finally, we consider applications of these methods to option pricing.

We consider a critical bisexual branching process in a random environment generated by independent and identically distributed random variables. Assuming that the process starts with a large number of pairs N, we prove that its extinction time is of order $\ln^2 N$. Interestingly, this result is valid for a general class of mating functions. Among these are the functions describing the monogamous and polygamous behavior of couples, as well as the function reducing the bisexual branching process to the simple one.

We consider a Lévy process reflected at the origin with additional independent and identically distributed collapses that occur at Poisson epochs, where a collapse is a jump downward to a state which is a random fraction of the state just before the jump. We first study the general case, then specialize to the case where the Lévy process is spectrally positive, and, finally, we specialize further to the two cases where the Lévy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.

We study a stochastic control problem where the underlying process follows a spectrally negative Lévy process. A controller can continuously increase the process but only decrease it at independent Poisson arrival times. We show the optimality of the periodic–classical barrier strategy, which increases the process whenever it would fall below some lower barrier and decreases it whenever it is observed above a higher barrier. An optimal strategy and the value function are written semi-explicitly using scale functions. Numerical results are also given.

We consider the problem of maximizing the expected average reward obtained over an infinite time horizon by n weakly coupled Markov decision processes. Our setup is a substantial generalization of the multi-armed restless bandit problem that allows for multiple actions and constraints. We establish a connection with a deterministic and continuous-variable control problem where the objective is to maximize the average reward derived from an occupancy measure that represents the empirical distribution of the processes when $n \to \infty$. We show that a solution of this fluid problem can be used to construct policies for the weakly coupled processes that achieve the maximum expected average reward as $n \to \infty$, and we give sufficient conditions for the existence of solutions. Under certain assumptions on the constraints, we prove that these conditions are automatically satisfied if the unconstrained single-process problem admits a suitable unichain and aperiodic policy. In particular, the assumptions include multi-armed restless bandits and a broad class of problems with multiple actions and inequality constraints. Also, the policies can be constructed in an explicit way in these cases. Our theoretical results are complemented by several concrete examples and numerical experiments, which include multichain setups that are covered by the theoretical results.

We study a family of Crump–Mode–Jagers branching processes in a random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of Iyer and the author (‘On the structure of genealogical trees associated with explosive Crump–Mode–Jagers branching processes’, arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of Iyer and the author and study a large range of previously unattainable cases.

In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.

We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the $L^1$ diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.

In this paper we propose a new efficient algorithm to compute the value function for zero-sum stopping games featuring two players with opposing interests. This can be seen as a game version of the ‘forward algorithm’ for (one-player) optimal stopping problems, first introduced by Irle (2006) for discrete-time Markov chains and later revisited by Miclo and Villeneuve (2021) for continuous-time Markov processes on general state spaces. This paper focuses on a game driven by a homogeneous continuous-time Markov chain taking values in a finite state space and also discusses the number of iterations needed. Illustrated computational implementations for a few particular examples are also provided.