We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433–443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156–180] studied the case when the risk process is modeled by a general spectrally negative Lévy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process.

We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function.

The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process.

One interpretation of the solution is that it is optimal for the risk-averse agent to gamble.

In this paper, in the particular case of a concave flux function, we are interested in the long time behavior of the nonlinear process associated in [Methodol. Comput. Appl. Probab. 2 (2000) 69–91] to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by replacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cumulative distribution function. We first obtain a trajectorial propagation of chaos estimate which strengthens the weak convergence result obtained in [8] without any convexity assumption on the flux function. Then Poincaré inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system.

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black–Scholes market considering Lipschitz continuous path-dependent payoffs for both discrete- and continuous-time cases. These results are new also for usual American style options. The paper continues and extends the study of Kifer [Ann. Appl. Probab. 16 (2006) 984–1033] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006) 984–1033] and the existence of optimal shortfall hedging in the discrete time established by Dolinsky and Kifer [Stochastics 79 (2007) 169–195].

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself.

In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.

The Poisson–Dirichlet distribution arises in many different areas. The parameter θ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of θ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson–Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter θ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson–Dirichlet distribution for large θ, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results.

We prove that any Brownian moving average

X_t=∫_−∞^t (f(s−t)−f(s)) dB_s, t≥0,

satisfies the conditional full support condition introduced by Guasoni, Rásonyi and Schachermayer [Ann. Appl. Probab. 18 (2008) 491–520].

Consider additive functionals of a Markov chain W_k, with stationary (marginal) distribution and transition function denoted by π and Q, say S_n=g(W₁)+⋯+g(W_n), where g is square integrable and has mean 0 with respect to π. If S_n has the form S_n=M_n+R_n, where M_n is a square integrable martingale with stationary increments and E(R_n²)=o(n), then g is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are E[E(S_n|W₁)²]=o(n) and lim_n→∞ E(S_n²)/n<∞. Assuming the first of these, let ‖g‖₊²=lim sup_n→∞ E(S_n²)/n; then ‖⋅‖₊ defines a pseudo norm on the subspace of L²(π) where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of ‖⋅‖₊. Let Q^* denote the adjoint operator to Q, regarded as a linear operator from L²(π) into itself, and consider co-isometries (QQ^*=I), an important special case that includes shift processes. In another main result a convenient orthonormal basis for L₀²(π) is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of g with respect to this basis.

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.

We develop a class of pathwise inequalities of the form H(B_t)≥M_t+F(L_t), where B_t is Brownian motion, L_t its local time at zero and M_t a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois’ Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _{0}^{\tau}\beta(B_{s})\,ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

Counter to the general notion that the regular tree is the worst case for decay of correlation between sets and nodes, we produce an example of a multi-spin interacting system which has uniqueness on the d-regular tree but does not have uniqueness on some infinite d-regular graphs.

A popular Bayesian nonparametric approach to survival analysis consists in modeling hazard rates as kernel mixtures driven by a completely random measure. In this paper we derive asymptotic results for linear and quadratic functionals of such random hazard rates. In particular, we prove central limit theorems for the cumulative hazard function and for the path-second moment and path-variance of the hazard rate. Our techniques are based on recently established criteria for the weak convergence of single and double stochastic integrals with respect to Poisson random measures. The findings are illustrated by considering specific models involving kernels and random measures commonly exploited in practice. Our abstract results are of independent theoretical interest and can be applied to other areas dealing with Lévy moving average processes. The strictly Bayesian analysis is further explored in a companion paper, where our results are extended to accommodate posterior analysis.

We consider a wide class of ergodic first passage percolation processes on ℤ² and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four-color Richardson’s growth model. This improves earlier results of Häggström and Pemantle [J. Appl. Probab. 35 (1995) 683–692], Garet and Marchand [Ann. Appl. Probab. 15 (2005) 298–330] and Hoffman [Ann. Appl. Probab. 15 (2005) 739–747] who proved that first passage percolation has at least two geodesics and that coexistence is possible in a two-color Richardson’s growth model.

A binary renewal process is a stochastic process {X_n} taking values in {0, 1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X₀, X₁, …, X_n one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.

The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability.

For a uniformly chosen random hypergraph of m=nρ vertices and n hyperedges, each consisting of the same fixed number l≥3 of vertices, the size of the core exhibits for large n a first-order phase transition, changing from o(n) for ρ>ρ_c to a positive fraction of n for ρ<ρ_c, with a transition window size Θ(n^−1/2) around ρ_c>0. Analyzing the corresponding “leaf removal” algorithm, we determine the associated finite-size scaling behavior. In particular, if ρ is inside the scaling window (more precisely, ρ=ρ_c+rn^−1/2), the probability of having a core of size Θ(n) has a limit strictly between 0 and 1, and a leading correction of order Θ(n^−1/6). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. This behavior is expected to be universal for a wide collection of combinatorial problems.

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance.