In this paper, we investigate a competitive market involving two agents who consider both their own wealth and the wealth gap with their opponent. Both agents can invest in a financial market consisting of a risk-free asset and a risky asset, under conditions where model parameters are partially or completely unknown. This setup gives rise to a nonzero-sum differential game within the framework of reinforcement learning (RL). Each agent aims to maximize his own Choquet-regularized, time-inconsistent mean-variance objective. Adopting the dynamic programming approach, we derive a time-consistent Nash equilibrium strategy in a general incomplete market setting. Under the additional assumption of a Gaussian mean return model, we obtain an explicit analytical solution, which facilitates the development of a practical RL algorithm. Notably, the proposed algorithm achieves uniform convergence, even though the conventional policy improvement theorem does not apply to the equilibrium policy. Numerical experiments demonstrate the robustness and effectiveness of the algorithm, underscoring its potential for practical implementation.

Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting. The core of this work is to generalise the discrete-time picture to continuous time by providing a notion of behavioural equivalence for continuous-time Markov processes. In Chen et al. [(2019). Electronic Notes in Theoretical Computer Science 347 45–63.], we proposed two equivalent definitions of bisimulation for continuous-time stochastic processes where the evolution is a flow through time: the first one as an equivalence relation and the second one as a cospan of morphisms. In Chen et al. [(2020). Electronic Notes in Theoretical Computer Science.], we developed the theory further: we introduced different concepts that correspond to different behavioural equivalences and compared them to bisimulation. In particular, we studied the relation between bisimulation and symmetry groups of the dynamics. We also provided a game interpretation for two of the behavioural equivalences. The present work unifies the cited conference presentations and gives detailed proofs.

The Drake equation has proven fertile ground for speculation about the abundance, or lack thereof, of communicating extraterrestrial intelligences (CETIs) for decades. It has been augmented by subsequent authors to include random variables in order to understand its probabilistic behaviour. However, in most cases, the emergence and lifetime of CETIs are assumed to be independent of each other. In this paper, we will derive several expressions that can demonstrate how CETIs may relate to each other in technological age as well as how the dynamics of the concurrent CETI population change under basic models of interaction, such as the Allee effect. By defining interaction as the change in the expected communication lifetime with respect to the density of CETI in a region of space, we can use models and simulation to understand how the CETI density can promote or inhibit the longevity and overall population of interstellar technological civilizations.

We describe a process where two types of particles, marked red and blue, arrive in a domain at a constant rate. When a new particle arrives into the domain, if there are particles of the opposite color present within a distance of 1 from the new particle, then, among these particles, it matches to the one with the earliest arrival time, and both particles are removed. Otherwise, the particle is simply added to the system. Additionally, particles may lose patience and depart at a constant rate. We study the existence of a stationary regime for this process, when the domain is either a compact space or a Euclidean space. In the compact setting, we give a product-form characterization of the stationary distribution, and then prove an FKG-type inequality that establishes certain clustering properties of the particles in the steady state.

Some annual weeds, perennial herbs, and suffrutescent half shrubs are unique in that their populations are cyclic in nature and relatively short-lived. This variability presents a confounding factor that makes management decisions about control of these weeds difficult. This paper presents a procedure, using Markov processes, that gives additional information for managing cyclic weed populations. The Markov chain model is used to evaluate the interyear patterns of broom snakeweed survival when growing on New Mexico rangeland.

Ant gardens (AGs) are specialized ant-plant associations where arboreal ants build their carton nests in association with epiphytes that use the carton as a substrate. Most of the epiphytes are planted by ants; therefore, seed selection by ants is a key driver of the epiphyte composition of AGs. However, deterministic post-dispersal factors, such as the surrounding environmental conditions and plant succession, may also influence epiphyte composition. Here we ask whether epiphyte composition on a local scale is associated with dispersal constraints, local environmental conditions (light availability, number of branches and nest height) or AG successional stage. We sampled all epiphyte species in 18 AGs formed by Camponotus femoratus and Crematogaster levior in Central Amazon, Brazil. AGs were located within a range of 1 km and at a maximum of 20 m from the edges of a dirt road within a primary forest. Epiphytic composition showed strong spatial structure, decreasing in similarity with increasing distance. Environmental conditions and AG successional stage were not related to AG floristic composition, suggesting a key role of stochastic processes related to seed dispersal. A combination of seed abundance and attractiveness in neighbouring AGs seems to drive the higher similarity in epiphyte composition among closer AGs.

In various biological systems and small scale technological applications particlestransiently bind to a cylindrical surface. Upon unbinding the particles diffuse in thevicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise toan effective surface translation, for which we here derive and discuss the dynamicequations, including additional surface diffusion. We discuss the time evolution of thenumber of surface-bound particles, the effective surface mean squared displacement, andthe surface propagator. In particular, we observe sub- and superdiffusive regimes. Aplateau of the surface mean-squared displacement reflects a stalling of the surfacediffusion at longer times. Finally, the corresponding first passage problem for thecylindrical geometry is analysed.

A stochastic simulation model of the transmission and maintenance of genetic heterogeneity in the absence and presence of external selection pressures is presented for polygamous intestinal helminths such as Ascaris. The model assumes that the density distribution of the adult parasites is highly aggregated and that density-dependent effects on fecundity are important. The model gives rise to stable infection rates in the host. Where the parasite population contains genetic heterogeneity, with the exception of stochastic fluctuations which models genetic drift, the ratio of the different alleles remained constant over extended periods of time. This result contrasts with that of an earlier analytical model (Anderson, R. M., May, M. R. & Gutpa S. (1989) Parasitology 99, S59–S79), in which uneven mating probabilities for the different combinations of worm possible in a host was postulated to inevitably lead to fixation of the most abundant allele. New results suggest that in spite of the restricted choice of mating available to a worm in the confines of a host, selection pressure always leads to enrichment of the parasites carrying resistant alleles.

A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.

In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a nonstationary process. To overcome this problem, we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study the ℒ2-convergence, nondegeneracy, and continuity of the limit process. Establishing a power law for its moments, we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.

In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.