A new non-conservative stochastic reaction–diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy. A local central limit theorem for reflected diffusions in bounded Lipschitz domains is also established as a crucial tool.

We consider a Bradley–Terry model in random environment where each player faces each other once. More precisely, the strengths of the players are assumed to be random and we study the influence of their distributions on the asymptotic number of potential winners. First, we prove that under moment and convexity conditions, the asymptotic probability that the best player wins is 1. The convexity condition is natural when the distribution of strengths is unbounded and, in the bounded case, when this convexity condition fails the number of potential winners grows at a rate depending on the tail of the distribution. We also study the minimal strength required for an additional player to win in this last case.

Let $\sqrt{N}+\lambda_{\max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the “real Ginibre matrix”). We study the large deviations behaviour of the limiting $N\rightarrow \infty $ distribution $\mathbb{P}[\lambda_{\max}<t]$ of the shifted maximal real eigenvalue $\lambda_{\max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, \begin{equation*}\mathbb{P}[\lambda_{\max}<t]=1-\frac{1}{4}\operatorname{erfc}(t)+O(e^{-2t^{2}}).\end{equation*} This is a rigorous confirmation of the corresponding result of [Phys. Rev. Lett. 99 (2007) 050603]. We also prove that the left tail is exponential, with correct asymptotics up to $O(1)$: for $t<0$, \begin{equation*}\mathbb{P}[\lambda_{\max}<t]=e^{\frac{1}{2\sqrt{2\pi }}\zeta (\frac{3}{2})t+O(1)},\end{equation*} where $\zeta $ is the Riemann zeta-function.

Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABMs) with the step initial condition; see [Garrod, Poplavskyi, Tribe and Zaboronski (2015)]. Therefore, the tail behaviour of the distribution of $X_{s}^{(\max)}$—the position of the rightmost annihilating particle at fixed time $s>0$—can be read off from the corresponding answers for $\lambda_{\max}$ using $X_{s}^{(\max)}\stackrel{D}{=}\sqrt{4s}\lambda_{\max}$.

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Itô process. As a consequence, we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Itô process in a one-sided manner without reflection.

In a model-free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semistatic strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $\omega \in \Omega$, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.

The Wright–Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article, we demonstrate that it is in fact possible to simulate exactly from a broad class of Wright–Fisher diffusion processes and their bridges. For those diffusions corresponding to reversible, neutral evolution, our key idea is to exploit an eigenfunction expansion of the transition function; this approach even applies to its infinite-dimensional analogue, the Fleming–Viot process. We then develop an exact rejection algorithm for processes with more general drift functions, including those modelling natural selection, using ideas from retrospective simulation. Our approach also yields methods for exact simulation of the moment dual of the Wright–Fisher diffusion, the ancestral process of an infinite-leaf Kingman coalescent tree. We believe our new perspective on diffusion simulation holds promise for other models admitting a transition eigenfunction expansion.

We prove that the linear statistics of the eigenvalues of a Wigner matrix converge to a universal Gaussian process on all mesoscopic spectral scales, that is, scales larger than the typical eigenvalue spacing and smaller than the global extent of the spectrum.

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^{d}$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance. A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high-dimensional setting. These bounds improve and extend the results of Dalalyan [J. R. Stat. Soc. Ser. B. Stat. Methodol. (2017) 79 651–676].

Kesten and Lee [Ann. Appl. Probab. 6 (1996) 495–527] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^{d}$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative? Error estimates in central limit theorems satisfied by many other standard functionals studied in geometric probability are known, but techniques employed to tackle the problem for those functionals do not apply directly to the minimal spanning tree. Thus, the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees has remained open. In this work, we establish bounds on the convergence rate for the Poissonized version of this problem by using a variation of Stein’s method. We also derive bounds on the convergence rate for the analogous problem in the setup of the lattice $\mathbb{Z}^{d}$.

The contribution of this paper is twofold. First, we develop a general technique to compute convergence rates in central limit theorems satisfied by minimal spanning trees on sequences of weighted graphs, including minimal spanning trees on Poisson points inside a sequence of growing cubes. Second, we present a way of quantifying the Burton–Keane argument for the uniqueness of the infinite open cluster. The latter is interesting in its own right and based on a generalization of our technique, Duminil-Copin, Ioffe and Velenik [Ann. Probab. 44 (2016) 3335–3356] have recently obtained bounds on probability of two-arm events in a broad class of translation-invariant percolation models.

By fixing all parameters in a phylogenetic likelihood model except for one branch length, one obtains a one-dimensional likelihood function. In this work, we introduce a mathematical framework to characterize the shapes of such one-dimensional phylogenetic likelihood functions. This framework is based on analyses of algebraic structures on the space of all frequency patterns with respect to a polynomial representation of the likelihood functions. Using this framework, we provide conditions under which the one-dimensional phylogenetic likelihood functions are guaranteed to have at most one stationary point, and this point is the maximum likelihood branch length. These conditions are satisfied by common simple models including all binary models, the Jukes–Cantor model and the Felsenstein 1981 model.

We then prove that for the simplest model that does not satisfy our conditions, namely, the Kimura 2-parameter model, the one-dimensional likelihood functions may have multiple stationary points. As a proof of concept, we construct a nondegenerate example in which the phylogenetic likelihood function has two local maxima and a local minimum. To construct such examples, we derive a general method of constructing a tree and sequence data with a specified frequency pattern at the root. We then extend the result to prove that the space of all rescaled and translated one-dimensional phylogenetic likelihood functions under the Kimura 2-parameter model is dense in the space of all nonnegative continuous functions on $[0,\infty)$ with finite limits. These results indicate that one-dimensional likelihood functions under advanced evolutionary models can be more complex than it is typically assumed by phylogenetic inference algorithms; however, these complexities can be effectively captured by the Kimura 2-parameter model.

Consider a Boolean model in $\mathbb{R}^{d}$ with balls of random, bounded radii with distribution $F_{0}$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_{\infty}$ is given by $\theta_{L}(t)=\mathbb{P}\{Z_{\infty}\cap L\ne\varnothing \}$, defined for each compact $L\subset\mathbb{R}^{d}$.

We prove for any fixed $L$ and $F_{0}$ that $\theta_{L}(t)$ is infinitely differentiable in $t$, except at the critical value $t_{c}$; we give a Margulis–Russo-type formula for the derivatives. More generally, allowing the distribution $F_{0}$ to vary and viewing $\theta_{L}$ as a function of the measure $F:=tF_{0}$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval.

We also prove that $\theta_{L}(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $\beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of Tanemura [J. Appl. Probab. 30 (1993) 382–396], on regularity of the supercritical Boolean model in $d\geq3$ with fixed-radius balls, to the case with bounded random radii.

We analyze a robust version of the Dynkin game over a set $\mathcal{P}$ of mutually singular probabilities. We first prove that conservative player’s lower and upper value coincide (let us denote the value by $V$). Such a result connects the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$ up to the first time $\tau_{*}$ when $V$ meets the lower payoff process $L$. If the probability set $\mathcal{P}$ is weakly compact, one can even find an optimal triplet $(\mathbb{P}_{*},\tau_{*},\gamma_{*})$ for the value $V_{0}$.

The mutual singularity of probabilities in $\mathcal{P}$ causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.

We extend the results of Arguin et al. [Probab. Theory Related Fields 157 (2013) 535–574] and Aïdékon et al. [Probab. Theory Related Fields 157 (2013) 405–451] on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the “location” of the particle in the underlying Galton–Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_{+}\times\mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_{+}\times\mathbb{R}$ with a random intensity measure $Z(dz)\times C\mathrm{e}^{-\sqrt{2}x}\,dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor [Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field (2016)].

We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semimartingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well-defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to large deviations theory, and in particular, the celebrated Gärtner–Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black–Scholes–Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.

This paper establishes conditions for optimality of an $(s,S)$ ordering policy for the minimization of the long-term average cost of one-dimensional diffusion inventory models. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Characterization of the cost of a general $(s,S)$ policy as a function $F$ of two variables naturally leads to a nonlinear optimization problem over the ordering levels $s$ and $S$. Existence of an optimizing pair $(s_{*},S_{*})$ is established for these models. Using the minimal value $F_{*}$ of $F$, along with $(s_{*},S_{*})$, a function $G$ is identified which is proven to be a solution of a quasi-variational inequality provided a simple condition holds. At this level of generality, optimality of the $(s_{*},S_{*})$ ordering policy is established within a large class of ordering policies for which local martingale and transversality conditions involving $G$ hold. For specific models, optimality of an $(s,S)$ policy in the general class of admissible policies can be established using comparison results. This most general optimality result is shown for the classical drifted Brownian motion inventory model with holding and fixed plus proportional ordering costs. Optimality of an $(s,S)$ ordering policy is also extended to the general class of admissible policies for a geometric Brownian motion inventory model with fixed plus level-dependent ordering costs. However, for a drifted Brownian motion process with reflection at $\{0\}$, a new class of non-Markovian policies is introduced which have lower costs than the $(s,S)$ policies. In addition, interpreting reflection at $\{0\}$ as “just-in-time” ordering, a necessary and sufficient condition is given that determines when just-in-time ordering is better than traditional $(s,S)$ policies.

We examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper, we handle the case of transient walks.

The two-parameter Poisson–Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman’s one-parameter Poisson–Dirichlet distribution and to certain Fleming–Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here, we shed some light on these dynamics by formulating a $K$-allele Wright–Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $N\to\infty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson–Dirichlet diffusion as $K\to\infty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove a priori that in the limit there is no “loss of mass”, that is, that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.