We present a closed-form solution to a discounted optimal stopping zero-sum game in a model based on a generalised geometric Brownian motion with coefficients depending on its running maximum and minimum processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the original process hits certain boundaries depending on the running values of the associated maximum and minimum processes. The proof is based on the reduction of the original game to the equivalent coupled free-boundary problem and the solution of the latter problem by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are partially determined as either unique solutions to the appropriate system of arithmetic equations or unique solutions to the appropriate first-order nonlinear ordinary differential equations. The results obtained are related to the valuation of the perpetual lookback game options with floating strikes in the appropriate diffusion-type extension of the Black–Merton–Scholes model.

We derive some key extremal features for stationary kth-order Markov chains that can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a threshold, as the threshold tends to the upper endpoint of the distribution. Unlike previous studies with $k>1$, we consider processes where standard limit theory describes each extreme event as a single observation without any information about the transition to and from the body of the distribution. Our work uses different asymptotic theory which results in non-degenerate limit laws for such processes. We study the extremal properties of the initial distribution and the transition probability kernel of the Markov chain under weak assumptions for broad classes of extremal dependence structures that cover both asymptotically dependent and asymptotically independent Markov chains. For chains with $k>1$, the transition of the chain away from the exceedance involves novel functions of the k previous states, in comparison to just the single value, when $k=1$. This leads to an increase in the complexity of determining the form of this class of functions, their properties, and the method of their derivation in applications. We find that it is possible to derive an affine normalization, dependent on the threshold excess, such that non-degenerate limiting behaviour of the process, in the neighbourhood of the threshold excess, is assured for all lags. We find that these normalization functions have an attractive structure that has parallels to the Yule–Walker equations. Furthermore, the limiting process is always linear in the innovations. We illustrate the results with the study of kth-order stationary Markov chains with exponential margins based on widely studied families of copula dependence structures.

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in a time-inhomogeneous random environment (RWRE), which is related to the BRWRE by the many-to-one formula. The key step is to figure out Tanaka’s decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting in itself.

We introduce a modification of the generalized Pólya urn model containing two urns, and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$ added to the urns after n draws, where $J\in\mathbb{N}$. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$, properly normalized and centered, converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second-order behavior occurs at $\sqrt{\rho}$ and not at $\rho/2$, where $\rho$ is the dominant eigenvalue of the mean replacement matrix.

In this article, we give explicit bounds on the Wasserstein and Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard normal distribution, using the results of Last et al. (Prob. Theory Relat. Fields 165, 2016). Relying on the theory developed by Saulis and Statulevicius in Limit Theorems for Large Deviations (Kluwer, 1991) and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results of Hillairet et al. (ALEA 19, 2022) and Khabou et al. (J. Theoret. Prob. 37, 2024) regarding the normal approximation and those of Zhu (Statist. Prob. Lett. 83, 2013) for moderate deviations.

Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran–Kingman model. For the sharp-peak landscape, we perform several exact computations and derive several exact formulas. We also obtain an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and does not require a specific asymptotic expansion (such as sending the length of the macromolecules to $\infty$ or the mutation probability to 0). With the help of these formulas, we present an original proof for the well-known phenomenon of the error threshold. We recover the limiting quasispecies distribution in the long-chain regime. We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. Going beyond the sharp-peak landscape, we consider fitness landscapes having finitely many peaks and a plateau-type landscape. Finally, within this framework, we prove rigorously the possible occurrence of the survival of the flattest, a phenomenon which was previously discovered by Wilke et al. (Nature 412, 2001) and which has been investigated in several works (see e.g. Codoñer et al. (PLOS Pathogens 2, 2006), Franklin et al. (Artificial Life 25, 2019), Sardanyés et al. (J. Theoret. Biol. 250, 2008), and Tejero et al. (BMC Evolutionary Biol. 11, 2011)).

When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the $\chi^2$ metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.

We analyze the process M(t) representing the maximum of the one-dimensional telegraph process X(t) with exponentially distributed upward random times and generally distributed downward random times. The evolution of M(t) is governed by an alternating renewal of two phases: a rising phase R and a constant phase C. During a rising phase, X(t) moves upward, whereas, during a constant phase, it moves upward and downward, continuing to move until it attains the maximal level previously reached. Under some choices of the distribution of the downward times, we are able to determine the distribution of C, which allows us to obtain some bounds for the survival function of M(t). In the particular case of exponential downward random times, we derive an explicit expression for the survival function of M(t). Finally, the moments of the first passage time $\Theta_w$ of the process X(t) through a fixed boundary $w>0$ are analyzed.

We study a system consisting of n particles, moving forward in jumps on the real line. Each particle can make both independent jumps, whose sizes have some distribution, and ‘synchronization’ jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. The system state is the empirical distribution of particle locations. We consider the mean-field asymptotic regime where $n\to\infty$. We prove that $v_n$, the steady-state speed of advance of the particle system, converges, as $n\to\infty$, to a limit $v_{**}$ which can easily be found from a minimum speed selection principle. Also we prove that as $n\to\infty$, the system dynamics converges to that of a deterministic mean-field limit (MFL). We show that the average speed of advance of any MFL is lower-bounded by $v_{**}$, and the speed of a ‘benchmark’ MFL, resulting from all particles initially being co-located, is equal to $v_{**}$. In the special case of exponentially distributed independent jump sizes, we prove that a traveling-wave MFL with speed v exists if and only if $v\ge v_{**}$, with $v_{**}$ having a simple explicit form; we also show the existence of traveling waves for the modified systems with a left or right boundary moving at a constant speed v. We provide bounds on an MFL’s average speed of advance, depending on the right tail exponent of its initial state. We conjecture that these results for exponential jump sizes extend to general jump sizes.

In this paper, we consider a joint drift rate control and two-sided impulse control problem in which the system manager adjusts the drift rate as well as the instantaneous relocation for a Brownian motion, with the objective of minimizing the total average state-related cost and control cost. The system state can be negative. Assuming that instantaneous upward and downward relocations take a different cost structure, which consists of both a setup cost and a variable cost, we prove that the optimal control policy takes an $\left\{ {\!\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right),\!\left\{ {{\mu ^{\ast}}(x)\,:\,x \in [ {{s^{\ast}},{S^{\ast}}}]} \right\}} \right\}$ form. Specifically, the optimal impulse control policy is characterized by a quadruple $\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right)$, under which the system state will be immediately relocated upwardly to ${q^{\ast}}$ once it drops to ${s^{\ast}}$ and be immediately relocated downwardly to ${Q^{\ast}}$ once it rises to ${S^{\ast}}$; the optimal drift rate control policy will depend solely on the current system state, which is characterized by a function ${\mu ^{\ast}}\!\left( \cdot \right)$ for the system state staying in $[ {{s^{\ast}},{S^{\ast}}}]$. By analyzing an associated free boundary problem consisting of an ordinary differential equation and several free boundary conditions, we obtain these optimal policy parameters and show the optimality of the proposed policy using a lower-bound approach. Finally, we investigate the effect of the system parameters on the optimal policy parameters as well as the optimal system’s long-run average cost numerically.

In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

A new risk measure, the Lambda Value-at-Risk (VaR), was proposed from a theoretical point of view as a generalization of the ordinary VaR in the literature. Motivated by the recent developments in risk sharing problems for the VaR and other risk measures, we study the optimization of risk sharing for the Lambda VaR. Explicit formulas of the inf-convolution and sum-optimal allocations are obtained with respect to the left Lambda VaRs, the right Lambda VaRs, or a mixed collection of the left and right Lambda VaRs. The inf-convolution of Lambda VaRs constrained to comonotonic allocations is investigated. Explicit formula for worst-case Lambda VaRs under model uncertainty induced by likelihood ratios is also given.

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob. 29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul. 17, 2019).

In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.

It is known that the simple slice sampler has robust convergence properties; however, the class of problems where it can be implemented is limited. In contrast, we consider hybrid slice samplers which are easily implementable and where another Markov chain approximately samples the uniform distribution on each slice. Under appropriate assumptions on the Markov chain on the slice, we give a lower bound and an upper bound of the spectral gap of the hybrid slice sampler in terms of the spectral gap of the simple slice sampler. An immediate consequence of this is that the spectral gap and geometric ergodicity of the hybrid slice sampler can be concluded from the spectral gap and geometric ergodicity of the simple version, which is very well understood. These results indicate that robustness properties of the simple slice sampler are inherited by (appropriately designed) easily implementable hybrid versions. We apply the developed theory and analyze a number of specific algorithms, such as the stepping-out shrinkage slice sampling, hit-and-run slice sampling on a class of multivariate targets, and an easily implementable combination of both procedures on multidimensional bimodal densities.

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.