In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X\mathop{=}\limits^{d}\,g((\xi_{i},X_{i}),i\geq 1)$. Here (ξ_i) and g(⋅) are given and the X_i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(⋅) is essentially a “maximum” or “minimum” function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X_i, are the X_i measurable functions of the innovations process (ξ_i)?

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Itô and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuation problems for contingent claims with a recursive payoff structure.

A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate θ that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ=dX−θ(Z) dt+dL−dU, where X is a (0,σ) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier at Z=b, respectively. The cumulative cost process increases according to the differential relationship dξ=c(θ(Z)) dt+p dU, where c(⋅) is a nondecreasing cost of control and p>0 is a penalty rate associated with displacement at the upper boundary. The objective is to minimize long-run average cost. This problem is solved explicitly, which allows one to also solve the following, essentially equivalent formulation: minimize the long-run average cost of control subject to an upper bound constraint on the average rate at which U increases. The two special problem features that allow an explicit solution are the use of a long-run average cost criterion, as opposed to a discounted cost criterion, and the lack of state-related costs other than boundary displacement penalties. The application of this theory to power control in wireless communication is discussed.

Let E_i be a collection of i.i.d. exponential random variables. Bouchaud’s model on ℤ is a Markov chain X(t) whose transition rates are given by w_ij=νexp(−β((1−a)E_i−aE_j)) if i, j are neighbors in ℤ. We study the behavior of two correlation functions: ℙ[X(t_w+t)=X(t_w)] and ℙ[X(t')=X(t_w) ∀ t'∈[t_w,t_w+t]]. We prove the (sub)aging behavior of these functions when β>1 and a∈[0,1].

This paper considers Gaussian flows multiplexed in a queueing network. A single node being a useful but often incomplete setting, we examine more advanced models. We focus on a (two-node) tandem queue, fed by a large number of Gaussian inputs. With service rates and buffer sizes at both nodes scaled appropriately, Schilder’s sample-path large-deviations theorem can be applied to calculate the asymptotics of the overflow probability of the second queue. More specifically, we derive a lower bound on the exponential decay rate of this overflow probability and present an explicit condition for the lower bound to match the exact decay rate. Examples show that this condition holds for a broad range of frequently used Gaussian inputs. The last part of the paper concentrates on a model for a single node, equipped with a priority scheduling policy. We show that the analysis of the tandem queue directly carries over to this priority queueing system.

We prove a strong law of large numbers for the location of the second class particle in a totally asymmetric exclusion process when the process is started initially from a decreasing shock. This completes a study initiated in Ferrari and Kipnis [Ann. Inst. H. Poincaré Probab. Statist. 13 (1995) 143–154].

We introduce a bond portfolio management theory based on foundations similar to those of stock portfolio management. A general continuous-time zero-coupon market is considered. The problem of optimal portfolios of zero-coupon bonds is solved for general utility functions, under a condition of no-arbitrage in the zero-coupon market. A mutual fund theorem is proved, in the case of deterministic volatilities. Explicit expressions are given for the optimal solutions for several utility functions.

We study the behavior of an infinite system of ordinary differential equations modeling the dynamics of a metapopulation, a set of (discrete) populations subject to local catastrophes and connected via migration under a mean field rule; the local population dynamics follow a generalized logistic law. We find a threshold below which all the solutions tend to total extinction of the metapopulation, which is then the only equilibrium; above the threshold, there exists a unique equilibrium with positive population, which, under an additional assumption, is globally attractive. The proofs employ tools from the theories of Markov processes and of dynamical systems.

We consider the problem of optimal stopping for a one-dimensional diffusion process. Two classes of admissible stopping times are considered. The first class consists of all nonanticipating stopping times that take values in [0,∞], while the second class further restricts the set of allowed values to the discrete grid {nh:n=0,1,2,…,∞} for some parameter h>0. The value functions for the two problems are denoted by V(x) and V^h(x), respectively. We identify the rate of convergence of V^h(x) to V(x) and the rate of convergence of the stopping regions, and provide simple formulas for the rate coefficients.

We consider a discrete-time financial market model with finite time horizon and give conditions which guarantee the existence of an optimal strategy for the problem of maximizing expected terminal utility. Equivalent martingale measures are constructed using optimal strategies.

We extend the Longstaff–Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results obtained by Clément, Lamberton and Protter [Finance and Stochastics 6 (2002) 449–471].

Let U be a given function defined on ℝ^d and π(x) be a density function proportional to exp−U(x). The following diffusion X(t) is often used to sample from π(x), $$dX(t)=-\nabla U(X(t))\,dt+\sqrt{2}\,dW(t),\qquad X(0)=x_{0}.$$ To accelerate the convergence, a family of diffusions with π(x) as their common equilibrium is considered, $$dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr)\,dt+\sqrt{2}\,dW(t),\qquad X(0)=x_{0}.$$ Let L_C be the corresponding infinitesimal generator. The spectral gap of L_C in L²(π) (λ(C)), and the convergence exponent of X(t) to π in variational norm (ρ(C)), are used to describe the convergence rate, where

λ(C)=Sup {real part of μ:μ is in the spectrum of L_C, μ is not zero}, $$\rho(C)=\operatorname {Inf}\biggl\{\rho: \int \vert p(t,x,y)-\pi(y)\vert \,dy\le g(x)e^{\rho t}\biggr\}.$$ Roughly speaking, L_C is a perturbation of the self-adjoint L₀ by an antisymmetric operator C⋅∇, where C is weighted divergence free. We prove that λ(C)≤λ(0) and equality holds only in some rare situations. Furthermore, ρ(C)≤λ(C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.

The natural analogue for a Lévy process of Cramér’s estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Lévy process with finite negative mean which satisfies Cramér’s condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of “high excursions.”

We consider planar lattice walks that start from (0,0), remain in the first quadrant i,j≥0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties:

• they are counted by nice numbers [Kreweras, Cahiers du B.U.R.O 6 (1965) 5–105],

• the generating function of these numbers is algebraic [Gessel, J. Statist. Plann. Inference 14 (1986) 49–58],

• the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function [Flatto and Hahn, SIAM J. Appl. Math. 44 (1984) 1041–1053].

These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds.

Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

A summation test is proposed to determine admissible types of gap penalties for logarithmic growth of the local alignment score. We also define a converging sequence of log moment generating functions that provide the constants associated with the large deviation rate and logarithmic strong law of the local alignment score and the asymptotic number of matches in the optimal local alignment.

In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching process’ framework. This density-dependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathematically into a quadratic death rate. The logistic branching process, or LB-process, can thus be seen as (the mass of ) a fragmentation process (corresponding to the branching mechanism) combined with constant coagulation rate (the death rate is proportional to the number of possible coalescing pairs). In the continuous state-space setting, the LB-process is a time-changed (in Lamperti’s fashion) Ornstein–Uhlenbeck type process. We obtain similar results for both constructions: when natural deaths do not occur, the LB-process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution of a Riccati differential equation. We also show that the quadratic regulatory term allows the LB-process to start at infinity, despite the fact that births occur infinitely often as the initial state goes to ∞. This result can be viewed as an extension of the pure-death process starting from infinity associated to Kingman’s coalescent, when some independent fragmentation is added.

We investigate Δ_n, the distance between randomly selected pairs of nodes among n keys in a random trie, which is a kind of digital tree. Analytical techniques, such as the Mellin transform and an excursion between poissonization and depoissonization, capture small fluctuations in the mean and variance of these random distances. The mean increases logarithmically in the number of keys, but curiously enough the variance remains O(1), as n→∞. It is demonstrated that the centered random variable Δ_n^*=Δ_n−⌊2log₂n⌋ does not have a limit distribution, but rather oscillates between two distributions.

We derive sufficient conditions for subgeometric f-ergodicity of strongly Markovian processes. We first propose a criterion based on modulated moment of some delayed return-time to a petite set. We then formulate a criterion for polynomial f-ergodicity in terms of a drift condition on the generator. Applications to specific processes are considered, including Langevin tempered diffusions on ℝⁿ and storage models.