A complete solution is provided to the infinite-horizon, discounted problem of optimal consumption and investment in a market with one stock, one money market (sometimes called a "bond") and proportional transaction costs. The utility function may be of the form $c^p/p$, where $p < 0$ or $0 < p < 1$, or may be $\log c$. It is assumed that the interest rate for the money market is positive, the mean rate of return for the stock is larger than this interest rate, the stock volatility is positive and all these parameters are constant. The only other assumption is that the value function is finite; necessary conditions for this are given. In the Appendix (by S. Shreve), the sensitivity of the value function under the assumption $0 < p < 1$ is shown to be of the order of the transaction cost to the 2/3 power. This implies that the liquidity premium associated with small transaction costs is also of the order of the transaction cost to the 2/3 power. Because this power is less than 1, the marginal liquidity premium turns out to be infinite. The analysis of this paper and its Appendix relies on the concept of viscosity solutions to Hamilton-Jacobi-Bellman equations. A self-contained treatment of this subject, adequate for the present application, is provided.

A class of open first-in, first-out queueing networks is examined. Customers arrive according to a rate-1 Poisson process and wait at queues along their prescribed routes for exponential holding times, after which they exit from the system. Such a network can be chosen so that the sum of the mean service times at each queue is as small as desired. It is shown here that these networks are nevertheless unstable. Each such network will possess two customer types, which proceed along nearly parallel routes. Queues are visited sequentially, with each consisting of one relatively slow step and then several quick steps.

We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourier-series method; that is, the inversion formula is the Fourier series of a periodic function constructed by aliasing. This amounts to an application of the Poisson summation formula. By appropriately exponentially damping the given function, we control the aliasing error. We choose the periods of the multidimensional periodic function so that each infinite series is a finite sum of nearly alternating infinite series. Then we apply the Euler transformation to compute the infinite series from finitely many terms. The multidimensional inversion algorithm enables us, evidently for the first time, to calculate probability distributions quickly and accurately from several classical transforms in queueing theory. For example, we apply our algorithm to invert the two-dimensional transforms of the joint distribution of the duration of a busy period and the number served in that busy period, and the time-dependent transient queue-length and workload distributions in the M/G/1 queue. In other related work, we have applied the inversion algorithms here to calculate time-dependent distributions in the transient BMAP/G/1 queue (with a batch Markovian arrival process) and the piecewise-stationary $M_t/G_t/1$ queue.

A fleet of vehicles located at a common depot must serve customers located throughout the plane. Without loss of generality, the depot will be located at the origin. Each vehicle must start at the depot, travel in turn to each customer its serves and go back to the depot. Each vehicle can serve at most $k$ customers. The objective is to minimize the total distance traveled by the fleet. In our model, the customers $X_1, \ldots, X_n$ are independent and uniformly distributed over the unit disc. If $R'(X_1, \ldots, X_n)$ denotes the optimal solution with these customer locations, we show that with overwhelming probability we have $\big|R'(X_1, \ldots, X_n) - \frac{2}{k} \sum_{i \leq n} \|X_i\| - \xi \sqrt{n} big| \leq K(n \log n)^{1/3},$ where $\xi$ and $K$ are constants that depend on $k$ only.

We study a continuous time inventory process that is a reflection mapping of a semimartingale netput process. Inventory processes of this type include the workload process in queues, dam and storage processes (with perhaps pure jump Levy input), as well as processes arising in fluid models. We establish sufficient conditions on the netput ensuring that the steady-state inventory has finite moments of order $k \geq 1$, and derive explicit bounds for these moments. The sufficient conditions require that the netput have a negative (local) drift and that the (conditional) $(k + 1)$th moment of its increments be bounded.

Let $\{Z_n\}^\infty_0$ be a Galton-Watson branching process with offspring distribution $\{p_j\}^\infty_0$. We assume throughout that $p_0 = 0, p_j \neq 1$ for any $j \geq 1$ and $1 < m = \Sigma jp_j < \infty$. Let $W_n = Z_nm^{-m}$ and $W = \lim_nW_n$. In this paper we study the rates of convergence to zero as $n \rightarrow \infty$ of $P\big(\big|\frac{Z_{n+1}}{Z_n} - m\big| > \varepsilon\big),\quad P(|W_n - W| > \varepsilon)$, $P\big(\big|\frac{Z_{n+1}}{Z_n} - m \mid > \varepsilon\big|W \geq a\big)$ for $\varepsilon > 0$ and $a > 0$ under various moment conditions on $\{p_j\}$. It is shown that the rate for the first one is geometric if $p_1 > 0$ and supergeometric if $p_1 = 0$, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.

A general framework in an ordinal utility setting for the analysis of dynamic choice from a continuum of alternatives $E$ is proposed. The model is based on the theory of random utility maximization in continuous time. We work with superextremal processes $\mathbf{Y} = \{\mathbf{Y}_t, t \in (0,\infty)\}$, where $\mathbf{Y}_t = \{Y_t(\tau),\tau \in E\}$ is a random element of the space of upper semicontinuous functions on a compact metric space $E$. Here $Y_t(\tau)$ represents the utility at time $t$ for alternative $\tau \in E$. The choice process $\mathbf{M} = \{M_t, t \in (0,\infty)\}$, is studied, where $M_t$ is the set of utility maximizing alternatives at time $t$, that is, $M_t$ is the set of $\tau \in E$ at which the sample paths of $\mathbf{Y}_t$ on $E$ achieve their maximum. Independence properties of $\mathbf{Y}$ and $\mathbf{M}$ are developed, and general conditions for $\mathbf{M}$ to have the Markov property are described. An example of such conditions is that $\mathbf{Y}$ have max-stable marginals.

We consider the problem of sampling according to a distribution with log-concave density $F$ over a convex body $K \subseteq \mathbb{R}^n$. The sampling is done using a biased random walk, and we prove polynomial upper bounds on the time to get a sample point with distribution close to $F$.

We prove that a certain (centered unimodal) rearrangement of coefficients in the moving average initial input process maximizes the variance (energy density) of the limit distribution of the spatiotemporal random field solution of a nonlinear partial differential equation called Burgers' equation. Our proof is in the spirit of domination principles developed in the book by Kwapien and Woyczynski.

We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter, we establish a random version of the Perron-Frobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics.

Functionals $L$ on finite subsets $A$ of $\mathbb{R}^d$ are considered for which the value is the minimum total edge length among a class of graphs with vertex set equal to, or in some cases containing, $A$. Examples include minimal spanning trees, the traveling salesman problem, minimal matching and Steiner trees. Beardwood, Halton and Hammersley, and later Steele, have shown essentially that for $\{X_1, \ldots, X_n\}$ a uniform i.i.d. sample from $\lbrack 0,1 \rbrack^d, EL(\{X_1, \ldots, X_n\})/n^{(d-1)/d}$ converges to a finite constant. Here we bound the rate of this convergence, proving a conjecture of Beardwood, Halton and Hammersley.

By embedding in a suitable continuous-time process, we find a strong law for $h_n$, the height of a random binary pyramid of order $n$. We show that $h_n/\ln n$ converges almost surely to a constant limit and we determine that limit.

We consider Boolean models in $d$-dimensional Euclidean space. Each point of a stationary, ergodic point process is the center of a ball with random radius. In this way, the space is partitioned into an occupied and a vacant region. We are interested in the number of unbounded occupied or vacant components that can coexist. We show that under very general conditions on the distribution of the radius random variable, there can be at most one unbounded component of each type. In case the point process is Poisson, we obtain uniqueness of the unbounded components without imposing any condition at all. Although we do not prove the necessity of the conditions to prove uniqueness, we obtain examples of stationary, ergodic point processes where the unbounded components are not unique when the conditions are violated. Finally, we discuss more general random shapes than just balls which are centered at the points of the point process.