The signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard its expectation, the expected signature, as a path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions with motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.

Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

This study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserves in a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on the smoothness of the ruin probability as a function of the initial capital, and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival (as well as the ruin) probability is a solution of an ordinary differential equation of the fourth order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-sided jumps.

In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).

We study an N-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

The Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $V(n,x) = \sup_{\tau }\mathbb{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$ , where S is a random walk. We give a tight upper bound for V when S has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow–Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 489\,241$ .

A diffusion approximation to a risk process under dynamic proportional reinsurance is considered. The goal is to minimise the discounted time in drawdown; that is, the time where the distance of the present surplus to the running maximum is larger than a given level $d > 0$ . We calculate the value function and determine the optimal reinsurance strategy. We conclude that the drawdown measure stabilises process paths but has a drawback as it also prevents surpassing the initial maximum. That is, the insurer is, under the optimal strategy, not interested in any more profits. We therefore suggest using optimisation criteria that do not avoid future profits.

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

For the gambler’s ruin problem with two players starting with the same amount of money, we show the playing time is stochastically maximized when the games are fair.

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Signal-to-interference-plus-noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, independent and identically distributed, and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or more dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor $\gamma$ and the SINR threshold $\tau$ satisfy $\gamma \geq 1/(2\tau)$ , then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $P(u)\;:\!=\; \mathbb{P}\{\cap_{i=1}^n (\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u )\}$ , $u\rightarrow\infty$ , where $X_i(t)$ , $t\ge0$ , $i=1,2,\ldots,n$ , are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \ldots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$ , $a_i>0$ , $i=1,2,\ldots,n$ . Our result shows that the structure of the asymptotics of P(u) is determined by the signs of the drifts $c_i$ . We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.

We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.

For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

We consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) hold. We also carry out some numerical experiments that fail to disprove this conjecture. Importantly, modelling folklore has often treated MMPPs as “bursty”, and implicitly assumed that (III) holds. However, to the best of our knowledge, proving this is still an open problem.

An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

We present a new and straightforward algorithm that simulates exact sample paths for a generalized stress-release process. The computation of the exact law of the joint inter-arrival times is detailed and used to derive this algorithm. Furthermore, the martingale generator of the process is derived, and induces theoretical moments which generalize some results of [3] and are used to demonstrate the validity of our simulation algorithm.