We introduce a financial market model featuring a risky asset whose price follows a sticky geometric Brownian motion and a riskless asset that grows with a constant interest rate $r\in \mathbb R$. We prove that this model satisfies no arbitrage and no free lunch with vanishing risk only when $r=0$. Under this condition, we derive the corresponding arbitrage-free pricing equation, assess the replicability, and give a representation of the replication strategy. We then show that all locally bounded replicable payoffs for the standard Black–Scholes model are also replicable for the sticky model. Last, we evaluate via numerical experiments the impact of hedging in discrete time and of misrepresenting price stickiness.

This paper focuses on the averaging principle concerning the fast–slow McKean–Vlasov stochastic differential equations driven by mixed fractional Brownian motion with Hurst parameter $\tfrac{1}{2} < H < 1$. The integral associated with Brownian motion is the standard Itô integral, while the integral with respect to fractional Brownian motion is a generalized Riemann–Stieltjes integral. Under the non-Lipschitz condition and certain appropriate assumptions regarding the coefficients, we initially establish the existence and uniqueness theorem for the fast–slow McKean–Vlasov stochastic differential equation driven by mixed fractional Brownian motion. Subsequently, we demonstrate the averaging principle of the fast–slow McKean–Vlasov stochastic differential equations, signifying that the slow stochastic differential equation converges to the associated averaged equation in terms of mean-square convergence.

We consider the problem of estimating fractional processes based on noisy high-frequency data. Generalizing the idea of pre-averaging to a fractional setting, we exhibit a sequence of consistent estimators for the unknown parameters of interest by proving a law of large numbers for associated variation functionals. In contrast to the semimartingale setting, the optimal window size for pre-averaging depends on the unknown roughness parameter of the underlying process. We evaluate the performance of our estimators in a simulation study and use them to empirically verify Kolmogorov’s $2/3$-law in turbulence data contaminated by instrument noise.

We investigate a specific class of irreducible, level-dependent, discrete-time, GI/M/1-type Markov chains. The transition matrices possess a block lower-Hessenberg structure, which shows asymptotic convergence along the rows as the level approaches infinity. Criteria are presented for recurrence, transience, positive recurrence, geometric ergodicity, and geometric transience in terms of elements of the transition matrices. These criteria are established by employing drift functions and matrix-generating functions. Furthermore, we discuss the extension of the main results to the continuous-time case.

We develop general conditions for weak convergence of adaptive Markov chain Monte Carlo processes and this is shown to imply a weak law of large numbers for bounded Lipschitz continuous functions. This allows an estimation theory for adaptive Markov chain Monte Carlo where previously developed theory in total variation may fail or be difficult to establish. Extensions of weak convergence to general Wasserstein distances are established, along with a weak law of large numbers for possibly unbounded Lipschitz functions. Applications are applied to autoregressive processes in various settings, unadjusted Langevin processes, and adaptive Metropolis–Hastings.

Consider nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,\ldots$ of a random variable X with a probability density function f. If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of $R=(X_{N+1},X_{N+2},\ldots)$ conditioned on $S=(X_1,\ldots,X_N)$ does not depend on f. If also the lengths of the intervals exhibit a Markovian structure, $R\mid S$ becomes a Markov chain of a certain order $s\ge0$. If $s=0$ then $X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time $\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.

We consider the problem of sequential matching in a stochastic block model with several classes of nodes and generic compatibility constraints. When the probabilities of connections do not scale with the size of the graph, we show that under the Ncond condition, a simple max-weight type policy allows us to attain an asymptotically perfect matching while no sequential algorithm attains perfect matching otherwise. The proof relies on a specific Markovian representation of the dynamics associated with Lyapunov techniques.

The Hawkes process is a popular candidate for researchers to model phenomena that exhibit a self-exciting nature. The classical Hawkes process assumes the excitation kernel takes an exponential form, thus suggesting that the peak excitation effect of an event is immediate and the excitation effect decays towards 0 exponentially. While the assumption of an exponential kernel makes it convenient for studying the asymptotic properties of the Hawkes process, it can be restrictive and unrealistic for modelling purposes. A variation on the classical Hawkes process is proposed where the exponential assumption on the kernel is replaced by integrability and smoothness type conditions. However, it is substantially more difficult to conduct asymptotic analysis under this setup since the intensity process is non-Markovian when the excitation kernel is non-exponential, rendering techniques for studying the asymptotics of Markov processes inappropriate. By considering the Hawkes process with a general excitation kernel as a stationary Poisson cluster process, the intensity process is shown to be ergodic. Furthermore, a parametric setup is considered, under which, by utilising the recently established ergodic property of the intensity process, consistency of the maximum likelihood estimator is demonstrated.

In this paper we derive cumulant bounds for subgraph counts and power-weighted edge lengths in a class of spatial random networks known as weight-dependent random connection models. These bounds give rise to different probabilistic results, from which we mainly focus on moderate deviations of the respective statistics, but also show a concentration inequality and a normal approximation result. This involves dealing with long-range spatial correlations induced by the profile function and the weight distribution. We start by deriving the bounds for the classical case of a Poisson vertex set, and then provide extensions to α-determinantal processes.

This work concerns stochastic differential equations with jumps. We prove convergence for solutions to a sequence of (possibly degenerate) stochastic differential equations with jumps when the coefficients converge in some appropriate sense. Then some special cases are analyzed and some concrete and verifiable conditions are given.

We study convergence rates, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact subset of $\mathbb{R}^d$. We propose two different approaches for this study. The first is based on the notion of a minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index, which is much used in extreme value theory. The second approach is based on a $\beta$-mixing condition together with a local-type dependence assumption. More precisely, all our results concern stationary $\beta$-mixing sequences satisfying a tail condition, known as the (a, b)-standard assumption, together with a local-type dependence condition or stationary sequences satisfying the (a, b)-standard assumption and having a positive minimal index. We prove that the optimal rates of the independent and identically distributed setting can be reached. We apply our results to stationary Markov chains on a ball, or to a class of Markov chains on a circle or on a torus. We study with simulations the particular examples of a Möbius Markov chain on the unit circle and of a Markov chain on the unit square wrapped on a torus.

We define a random graph obtained by connecting each point of $\mathbb{Z}^d$ independently and uniformly to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed k-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between the vertices in at least one, respectively precisely two, directions. For these graphs we study the question of percolation, i.e. the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected k-neighbor graph never percolates, while the directed k-neighbor graph percolates whenever $k \geq d+1$, $k \geq 3$, and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected 2-neighbor graph percolates for $d=2$, the undirected 3-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

We investigate geometric properties of invariant spatio-temporal random fields $X\colon\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space $\mathbb M^d$ in any dimension $d\ge 2$, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as $T\to +\infty$) of the average on [0,T] of the volume of the excursion set on the manifold, i.e. of $\lbrace X(\cdot, t)\ge u\rbrace$ (for any $u >0$). The Fourier components of X may have short or long memory in time, i.e. integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in Marinucci et al. (2021) and allows us to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as $T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field X.

The present paper develops a unified approach when dealing with short- or long-range dependent processes with finite or infinite variance. We are concerned with the convergence rate in the strong law of large numbers (SLLN). Our main result is a Marcinkiewicz–Zygmund law of large numbers for $S_{n}(f)= \sum_{i=1}^{n}f(X_{i})$, where $\{X_i\}_{i\geq 1}$ is a real stationary Gaussian sequence and $f(\!\cdot\!)$ is a measurable function. Key technical tools in the proofs are new maximal inequalities for partial sums, which may be useful in other problems. Our results are obtained by employing truncation alongside new maximal inequalities. The result can help to differentiate the effects of long memory and heavy tails on the convergence rate for limit theorems.

We consider a stochastic model, called the replicator coalescent, describing a system of blocks of k different types that undergo pairwise mergers at rates depending on the block types: with rate $C_{ij}\geq 0$ blocks of type i and j merge, resulting in a single block of type i. The replicator coalescent can be seen as a generalisation of Kingman’s coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.

The scale function plays a significant role in the fluctuation theory of Lévy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, which generally lacks an explicit representation. This paper introduces a novel series representation for the scale function, utilizing Laguerre polynomials to construct a uniformly convergent approximation sequence. Additionally, we conduct statistical inference based on specific discrete observations and propose estimators for the scale function that are asymptotically normal.

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes and, unlike previous results, they depend on the number of balls of each color in the urn. The connection of these results with finite de Finetti-style theorems is explored, and it is observed that a sampling bound due to Stam (1978) combined with the convexity of relative entropy yield a new finite de Finetti bound in relative entropy, which achieves the optimal asymptotic convergence rate.

The gambler’s ruin problem for correlated random walks (CRWs), both with and without delays, is addressed using the optional stopping theorem for martingales. We derive closed-form expressions for the ruin probabilities and the expected game duration for CRWs with increments $\{1,-1\}$ and for symmetric CRWs with increments $\{1,0,-1\}$ (CRWs with delays). Additionally, a martingale technique is developed for general CRWs with delays. The gambler’s ruin probability for a game involving bets on two arbitrary patterns is also examined.

In this paper we study the optimal multiple stopping problem with weak regularity for the reward, where the reward is given by a set of random variables indexed by stopping times. When the reward family is upper semicontinuous in expectation along stopping times, we construct the optimal multiple stopping strategy using the auxiliary optimal single stopping problems. We also obtain the corresponding results when the reward is given by a progressively measurable process.

The continuous random energy model (CREM) was introduced by Bovier and Kurkova in 2004 as a toy model of disordered systems. Among other things, their work indicates that there exists a critical point $\beta_\mathrm{c}$ such that the partition function exhibits a phase transition. The present work focuses on the high-temperature regime where $\beta<\beta_\mathrm{c}$. We show that, for all $\beta<\beta_\mathrm{c}$ and for all $s>0$, the negative s moment of the CREM partition function is comparable with the expectation of the CREM partition function to the power of $-s$, up to constants that are independent of N.