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 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 investigate the limiting spectral distribution of a noncentral unified matrix model defined by $\boldsymbol{\Omega}(\mathbf{X}) = ({(\mathbf{X}\mathbf{P}_1+\mathbf{A})(\mathbf{X}\mathbf{P}_1+\mathbf{A})'}/{n_1}) ({\mathbf{X}\mathbf{P}_2\mathbf{X}'}/{n_2})^{-1}$, where $\mathbf{X}=(X_{ij})_{p\times n}$ is a random matrix with independent and identically distributed real entries having zero mean and finite second moment. $\mathbf{A}$ is a $p\times n$ nonrandom matrix. The matrices $\mathbf{P}_1$ and $\mathbf{P}_2$ are projection matrices satisfying $\mathrm{rank}(\mathbf{P}_1)=n_1$, $\mathrm{rank}(\mathbf{P}_2)=n_2$, and $\mathbf{P}_1\mathbf{P}_2=0$. When $\mathbf{P}_1$ and $\mathbf{P}_2$ are random, they are assumed to be independent of $\mathbf{X}$. When $p/n_1\to c_1\in(0,\infty)$ and $p/n_2\to c_2\in(0,1)$, we establish the almost sure convergence of the empirical spectral distribution of $\boldsymbol{\Omega}$ to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.

We consider shock models governed by the bivariate geometric counting process. By assuming the competing risks framework, failures are due to one of two mutually exclusive causes (shocks). We obtain and study some relevant functions, such as failure densities, survival functions, probability of the cause of failure, and moments of the failure time conditioned on a specific cause. Such functions are specified by assuming that systems or living organisms fail at the first instant in which a random threshold is reached by the sum of received shocks. Under this failure scheme, various cases arising for suitable choices of the random threshold are provided too.

In recent works on the theory of machine learning, it has been observed that heavy tail properties of stochastic gradient descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, Gürbüzbalaban et al. (2021) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_n=A_nX_{n-1}+B_n$ for independent and identically distributed pairs $(A_n,B_n)$, where $A_n$ is a random symmetric matrix and $B_n$ is a random vector. However, their approach is not completely correct and, in the present paper, the problem is put into the right framework by applying the theory of irreducible-proximal matrices.

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.

We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied by Volkov (2009 ALEA 6, 399–414). Unlike that model, we allow delays in the spread of the fires and the non-zero burning time of individual ‘trees’. We obtain some general properties for this model, which cover, among others, the phenomenon of an ‘infinite fire’, not present in the original model.

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.

A random variable $\xi$ has a light-tailed distribution (for short, is light-tailed) if it possesses a finite exponential moment, ${\mathbb{E}} \, {\exp}{(\lambda \xi)} <\infty$ for some $\lambda >0$, and has a heavy-tailed distribution (is heavy-tailed) if ${\mathbb{E}} \, {\exp}{(\lambda\xi)} = \infty$ for all $\lambda>0$. Leipus et al. (2023 AIMS Math. 8, 13066–13072) presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. We show that this phenomenon is universal: any light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables. Moreover, a more general fact holds: these two independent random variables may have as heavy-tailed distributions as we wish. Further, we extend the latter result to the minimum of any finite number of independent random variables. We also comment on possible generalizations of our result to the case of dependent random variables.

We consider two-person zero-sum semi-Markov games with incomplete reward information on one side under the expected discount criterion. First, we prove that the value function exists and satisfies the Shapley equation. From the Shapley equation, we construct an optimal policy for the informed player. Second, to show the existence of an optimal policy for the uninformed player, we introduce an auxiliary dual game and establish the relationship between the primal game and the dual game. By this relationship, we also prove the existence of the value function of the dual game, and then construct an optimal policy for the uninformed player in the primal game. Finally, we develop two iterative algorithms to compute $\varepsilon$-optimal policies for the informed player and the uninformed player, respectively.

We investigate some investment problems related to maximizing the expected utility of the terminal wealth in a continuous-time Itô–Markov additive market. In this market, the prices of financial assets are described by Markov additive processes that combine Lévy processes with regime-switching models. We give explicit expressions for the solutions to the portfolio selection problem for the hyperbolic absolute risk aversion (HARA) utility, the exponential utility, and the extended logarithmic utility. In addition, we demonstrate that the solutions for the HARA utility are stable in terms of weak convergence when the parameters vary in a suitable way.

We study bond and site Bernoulli percolation models on $\mathbb{Z}^d$ for $d \geq 3$ with parameter p, in both the oriented and non-oriented versions. The main macroscopic quantity of interest is the probability of long-range order, and the existence of a non-trivial threshold is well established. Precise numerical results for the threshold values are available in the literature, but mathematically rigorous bounds are mostly restricted to two-dimensional lattices. Utilizing dynamical coupling techniques, we introduce a comprehensive set of new rigorous upper bounds that corroborate existing numerical values.

For Markov chains and Markov processes exhibiting a form of stochastic monotonicity (higher states have higher transition probabilities in terms of stochastic dominance), stability and ergodicity results can be obtained with the use of order-theoretic mixing conditions. We complement these results by providing quantitative bounds on deviations between distributions. We also show that well-known total variation bounds can be recovered as a special case.

Following the pivotal work of Sevastyanov (1957), who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process has been considered in depth. In this work, we demonstrate how we can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.

The normalised partial sums of values of a nonnegative multiplicative function over divisors with appropriately restricted lengths of a random permutation from the symmetric group define trajectories of a stochastic process. We prove a functional limit theorem in the Skorokhod space when the permutations are drawn uniformly at random. Furthermore, we show that the paths of the limit process almost surely belong to the space of continuous functions on the unit interval and, exploiting results from number-theoretic papers, we obtain rather complex formulas for the limits of joint power moments of the process.

Hybrid stochastic differential equations (SDEs) are a useful tool for modeling continuously varying stochastic systems modulated by a random environment, which may depend on the system state itself. In this paper we establish the pathwise convergence of solutions to hybrid SDEs using space-grid discretizations. Though time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov-modulated Brownian motions. This connection allows us to explore aspects that have been largely unexplored in the hybrid SDE literature. Specifically, we exploit our convergence result to obtain efficient and computationally tractable approximations for first-passage probabilities and expected occupation times of the solutions to hybrid SDEs. Lastly, we illustrate the effectiveness of the resulting approximations through numerical examples.

We study the bilateral preference graphs $\mathit{LK}(n, k)$ of La and Kabkab, obtained as follows. Put independent and uniform [0, 1] weights on the edges of the complete graph $K_n$. Then, each edge (i, j) is included in $\mathit{LK}(n,k)$ if it is bilaterally preferred, in the sense that it is among the k edges of lowest weight incident to vertex i, and among the k edges of lowest weight incident to vertex j. We show that $k = \log(n)$ is the connectivity threshold, solving a conjecture of La and Kabkab, and obtaining finer results about the window. We also investigate the asymptotic behavior of the average degree of vertices in $\mathit{LK}(n, k)$ as $n\rightarrow\infty$.

This paper considers option valuation under finite mixture models in a discrete-time economy. Specifically, the Esscher transform is employed to select a pricing kernel. Novel finite mixture models with negative-shifted Gamma and negative-shifted inverse Gaussian distributions are developed. A hybrid finite mixture model that allows different parametric forms for component distributions is introduced to incorporate model uncertainty. An empirical characteristic function estimation method is employed to estimate the finite mixture models. Closed-form pricing formulas for a European call option are obtained for some finite mixture models. Empirical examples using data on the Bitcoin-USD prices are provided to illustrate an application of the proposed models to value Bitcoin options.

General additive functionals of patricia tries are studied asymptotically in a probabilistic model with independent, identically distributed letters from a finite alphabet. Asymptotic normality is shown after normalization together with asymptotic expansions of the moments. There are two regimes depending on the algebraic structure of the letter probabilities, with and without oscillations in the expansion of moments. As applications firstly the proportion of fringe trees of patricia tries with k keys is studied, which is oscillating around $(1-\rho(k))/(2H)k(k-1)$, where H denotes the source entropy and $\rho(k)$ is exponentially decreasing. The oscillations are identified explicitly. Secondly, the independence number of patricia tries and of tries is considered. The general results for additive functions also apply, where a leading constant is numerically approximated. The results extend work of Janson on tries by relating additive functionals on patricia tries to additive functionals on tries.