We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of $\mathscr{R}^d$. The setting is that of a continuous-time, Ito process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory.

The multifractal structure of a measure refers to some notion of dimension of the set which supports singularities of a given order $\alpha$ as a function of the parameter $\alpha$. Measures with a nontrivial multifractal structure are commonly referred to as multifractals. Multifractal measures are being studied both empirically and theoretically within the statistical theory of turbulence and in the study of strange attractors of certain dynamical systems. Conventional wisdom suggests that various definitions of the multifractal structure of random cascades exist and coincide. While this is rigorously known to be the case for certain deterministic cascade measures, the same is not true for random cascades. The purpose of this paper is to pursue this theory for a class of random cascades. In addition to providing a new role for the modified cumulant generating function (structure function) studied by Mandelbrot, Kahane and Peyriere, the results have implications for the theoretical interpretation of empirical data on turbulence and rainfall distributions.

Consider any norm $N$ on $\mathbb{R}^d, d \geq 3$, and independent uniformly distributed points $X_1, \ldots, X_n, \ldots; Y_1, \ldots, Y_n, \ldots$ in $\lbrack 0, 1\rbrack^d$. Consider the random variable $M_n = \inf \sum_{i \leq n} N(X_i - Y_{\sigma(i)})$, where the infimum is taken over all permutations $\sigma$ of $\{1, \ldots, n\}$. We show that for some universal constant $K$, we have $\lim \sup_{n \rightarrow \infty} M_n n^{-1 + 1/d} \leq r_N \big(1 + K \frac{\log d}{d}\big)mathrm{a,s.},$ where $r_N$ is the radius of the ball for $N$ of volume 1.

Let $\{X_n\} = \{X^{(N)}_n\}$ be a Markov chain in the probability measures $\mathbf{P}\lbrack 0, 1\rbrack$, equipped with a certain metric for the topology of weak convergence, and denote $\mathbf{E}_x(X_1) = \mathbf{f}_N(x)$. Define a projection $\xi = \pi_dx$ on $\mathbf{P}\lbrack 0, 1\rbrack$ by defining $\xi$ to be absolutely continuous with respect to Lebesgue measure with constant density $(d + 1)x(A)$ on each interval $A$ of an equipartition of $\lbrack 0, 1 \rbrack$ into $d + 1$ intervals, and let $\pi_d\mathbf{P}\lbrack 0, 1\rbrack \subset \mathbb{R}^d$ be the natural embedding. Assume $\pi_d\mathbf{f}_N(\xi) \underline{\sim} \xi + \beta_N h_d(\xi), h_d(\xi_{0, d}) = \mathbf{0}$ and $\operatorname{Cov}_\xi(\pi_d X_1) \underline{\sim} \sigma^2_d(\xi)/N$, in certain senses as $N \rightarrow \infty$, where $\xi_{0, d}$ is an asymptotically stable fixed point of $\pi_d\mathbf{f}_N(\xi) = \xi$ and $x_0 = \lim_{d \rightarrow \infty} \xi_{0, d}$ exists in $\mathbf{P}\lbrack 0, 1\rbrack$. Assuming various regularity conditions and $\beta = \beta_N \rightarrow 0, N\beta/\log N \rightarrow \infty$, it is shown that the expected time it takes the Markov chain to exit a fixed open ball $D$ about $x_0$ once $X_0 \in D$ is logarithmically equivalent to $\exp\lbrack N\beta_N V \rbrack$, where $V > 0$ is a limit of solutions $V_d = V_d(h_d, \sigma_d)$ of variational problems of Wentzell-Freidlin type in $\mathbb{R}^d$ as $d \rightarrow \infty$. These results apply to an infinite alleles model in population genetics, where $\{X_n\}$ represents the evolution of distributions of types among a population of $N$ randomly mating genes, and where forces of mutation and selection are stronger than effects due to finite population size.

Consider a ring on which a server travels at constant speed. Customers arrive on the ring according to a Poisson process, at locations independently and uniformly distributed over the circle. Whenever the server encounters a customer, he stops and serves the client according to a general service time distribution. After the service is completed, the server removes the customer from the ring and resumes his round. The model is analyzed by means of point processes and regenerative processes in combination with some stochastic integration theory. This approach clarifies the analysis of the continuous polling model and provides the means for further generalizations. For every time $t$, the locations of customers that are waiting for service and the positions of clients that have been served during the last tour of the server are represented by random counting measures. These measures converge in distribution as $t \rightarrow \infty$. A recursive expression for the Laplace functionals of the limiting random measures is found, from which the corresponding $k$th moment measures can be derived.

A one-dimensional cellular automaton with conservative dynamics is studied. The automaton are a special case of the Boghosian-Levermore model, a probabilistic automaton, that has been used as a microscopic approximation for the Burgers equation. We study the stationary measures, the hydrodynamical limit and the existence of a microscopic interface.

We give a formula for the quadratic Lyapunov exponent of the harmonic oscillator in the presence of a finite-state Markov noise process. In case the noise process is reversible, the quadratic Lyapunov exponent is strictly less than that for the corresponding white-noise process obtained from the central limit theorem. An example is presented of a nonreversible Markov noise process for which this inequality is reversed.

When making probabilistic models for survival times, one should consider the fact that individuals are heterogeneous. The observed changes in population intensities (or hazard rates) over time are a mixed result of two influences: on the one hand, the actual changes in the individual hazards, and, on the other hand, the selection due to high-risk individuals leaving the risk group early. I will consider the common multiplicative model for heterogeneity, but with the new feature that the random proportionality factor has a compound Poisson distribution. This distribution is studied in some detail. It is pointed out how its application to the survival situation extends a model of Hougaard, inheriting several nice properties. One important feature of the model is that it yields a subgroup of zero susceptibility, which "survives forever." This is a relevant model in medicine and demography. Two examples are given where the model is fitted to data concerning marriage rates and fertility.

We consider a real-valued branching Brownian motion where particles are killed at rate $\mu$ and split at rate $\lambda \leq \mu$ into two independent offspring particles. The process dies out almost surely, so it reaches some lowest level. We prove a decomposition of the branching Brownian path at its minimum. The post-minimum path is just branching Brownian motion conditioned never to go beneath its initial point. The pre-minimum piece is independent of the post-minimum piece, and has the same law as the post-minimum piece reweighted by a functional of the endpoints of the tree. Applications to branching polymers are discussed.

Conditions for the finiteness of waiting-time moments in queues with a renewal arrival process were established by Kiefer and Wolfowitz. This paper establishes analogous conditions, some necessary, and some sufficient, in single-server queues with a general stationary ergodic arrival process. The feature of the arrival process in contributing to delay is any tendency to form clumps (or, clusters) of arrivals. In the more familiar setting of a renewal arrival process, the regenerative nature of the process severely limits any such tendency. More generally, strong mixing conditions on the sequence of interarrival times are used to give a sufficient condition for the finiteness of waiting-time moments. The details are worked out for the two important examples where the arrivals are generated by a Cox process and where the sequence of interarrival times contains an embedded stationary regenerative phenomenon. The latter example sheds light on the recent work of Wolff and the range of examples and counterexamples used to elaborate the theoretical results presented.

The problem of determining the length $L_n$ of the longest increasing subsequence in a random permutation of $\{1, \ldots, n\}$ is equivalent to that of finding the height of a random two-dimensional partial order (obtained by intersecting two random linear orders). The expectation of $L_n$ is known to be about $2\sqrt{n}$. Frieze investigated the concentration of $L_n$ about this mean, showing that, for $\varepsilon > 0$, there is some constant $\beta > 0$ such that $Pr(|L_n - \mathbf{E}L_n| \geq n^{1/3+\varepsilon}) \leq \exp(-n^\beta).$ In this paper we obtain similar concentration results for the heights of random $k$-dimensional orders, for all $k \geq 2$. In the case $k = 2$, our method replaces the $n^{1/3+\varepsilon}$ above with $n^{1/4+\varepsilon}$, which we believe to be essentially best possible.

This resolves a paradox in an application of a Markov chain model to market share estimation.

This paper considers the multiarmed bandit problem and presents a new proof of the optimality of the Gittins index policy. The proof is intuitive and does not require an interchange argument. The insight it affords is used to give a streamlined summary of previous research and to prove a new result: The optimal value function is a submodular set function of the available projects.