Motivated by applications to bond markets, we propose a multivariate framework for discrete time financial markets with proportional transaction costs and a countable infinite number of tradable assets. We show that the no-arbitrage of second kind property (NA2 in short), recently introduced by Rásonyi for finite-dimensional markets, allows us to provide a closure property for the set of attainable claims in a very natural way, under a suitable efficient friction condition. We also extend to this context the equivalence between NA2 and the existence of many (strictly) consistent price systems.

We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Itô-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system of quadratic backward stochastic differential equations (BSDEs) in Brownian filtration, and our main result proves, under fairly general conditions, the existence and uniqueness of a solution to this system, which characterizes explicitly the value function and optimal strategies to the optimal investment problem. We illustrate our solutions approach with some numerical tests emphasizing the impact of default intensities, loss or gain at defaults and correlation between assets. Beyond the financial problem, our decomposition approach provides a new perspective for solving quadratic BSDEs with a finite number of jumps.

The following distributed coalescence protocol was introduced by Dahlia Malkhi in 2006 motivated by applications in social networking. Initially there are $n$ agents wishing to coalesce into one cluster via a decentralized stochastic process, where each round is as follows: every cluster flips a fair coin to dictate whether it is to issue or accept requests in this round. Issuing a request amounts to contacting a cluster randomly chosen proportionally to its size. A cluster accepting requests is to select an incoming one uniformly (if there are such) and merge with that cluster. Empirical results by Fernandess and Malkhi suggested the protocol concludes in $O(\log n)$ rounds with high probability, whereas numerical estimates by Oded Schramm, based on an ingenious analytic approximation, suggested that the coalescence time should be super-logarithmic.

Our contribution is a rigorous study of the stochastic coalescence process with two consequences. First, we confirm that the above process indeed requires super-logarithmic time w.h.p., where the inefficient rounds are due to oversized clusters that occasionally develop. Second, we remedy this by showing that a simple modification produces an essentially optimal distributed protocol; if clusters favor their smallest incoming merge request then the process does terminate in $O(\log n)$ rounds w.h.p., and simulations show that the new protocol readily outperforms the original one. Our upper bound hinges on a potential function involving the logarithm of the number of clusters and the cluster-susceptibility, carefully chosen to form a supermartingale. The analysis of the lower bound builds upon the novel approach of Schramm which may find additional applications: rather than seeking a single parameter that controls the system behavior, instead one approximates the system by the Laplace transform of the entire cluster-size distribution.

A stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different time-scales, the subnetworks can be approximated separately, providing insight into the behavior of the full network through the analysis of these lower-dimensional approximations.

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halász. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants $H_{\alpha}$, as $\alpha\rightarrow0$.

Applying Stein’s method, an inductive technique and size bias coupling yields a Berry–Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdős–Rényi random graph of a given degree.

We study two of the simple rules on finite graphs under the death–birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman and Nowak [Nature 441 (2006) 502–505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.

We consider a utility-maximization problem in a general semimartingale financial model, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin; that is, it may be inadmissible for an investor to hold no risky investment at all. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present.

Our main result establishes the existence of optimal trading strategies in such models under no smoothness requirements on the utility function. The result also shows that, up to attainment, the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.

We present an efficient phylogenetic reconstruction algorithm allowing insertions and deletions which provably achieves a sequence-length requirement (or sample complexity) growing polynomially in the number of taxa. Our algorithm is distance-based, that is, it relies on pairwise sequence comparisons. More importantly, our approach largely bypasses the difficult problem of multiple sequence alignment.

Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis–Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an “infinite” degree component, that is, when an atypical “condensation” effect occurs with respect to the degree structure.

As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree distribution is deduced in terms of a coupled system of ODEs from which power law bounds for the limiting degree distribution are given. However, by analyzing the rate function, one can see that the process can deviate to a variety of atypical nonpower law distributions with finite cost, including distributions typically associated with sub and superlinear selection models.

The Markowitz problem consists of finding, in a financial market, a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: trading strategies must take values in a (possibly random and time-dependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in $L^{2}$. Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes $L^{\pm}$ appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of $L^{\pm}$ or equivalently into a coupled system of backward stochastic differential equations for $L^{\pm}$. We show how this can be used to both characterize and construct optimal strategies. Our results explain and generalize all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

We provide a nonasymptotic analysis of convergence to stationarity for a collection of Markov chains on multivariate state spaces, from arbitrary starting points, thereby generalizing results in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777]. Our examples include the multi-allele Moran model in population genetics and its variants in community ecology, a generalized Ehrenfest urn model and variants of the Pólya urn model. It is shown that all these Markov chains are stochastically monotone with respect to an appropriate partial ordering. Then, using a generalization of the results in [Diaconis, Khare and Saloff-Coste Sankhya 72 (2010) 45–76] and [Wilson Ann. Appl. Probab. 14 (2004) 274–325] (for univariate totally ordered spaces) to multivariate partially ordered spaces, we obtain explicit nonasymptotic bounds for the distance to stationarity from arbitrary starting points. In previous literature, bounds, if any, were available only from special starting points. The analysis also works for nonreversible Markov chains, and allows us to analyze cases of the multi-allele Moran model not considered in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777].

This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of multidimensional SDEs, and to find ways to “design” and optimize the distribution of the approximation error. As examples we briefly discuss strategies for discrete option hedging.