We discuss the problem of pricing contingent claims, such as European call options, based on the fundamental principle of "absence of arbitrage" and in the presence of constraints on portfolio choice, for example, incomplete markets and markets with short-selling constraints. Under such constraints, we show that there exists an arbitrage-free interval which contains the celebrated Black-Scholes price (corresponding to the unconstrained case); no price in the interior of this interval permits arbitrage, but every price outside the interval does. In the case of convex constraints, the endpoints of this interval are characterized in terms of auxiliary stochastic control problems, in the manner of Cvitanić and Karatzas. These characterizations lead to explicit computations, or bounds, in several interesting cases. Furthermore, a unique fair price $\hat{p}$ is selected inside this interval, based on utility maximization and "marginal rate of substitution" principles. Again, characterizations are provided for $\hat{p}$, and these lead to very explicit computations. All these results are also extended to treat the problem of pricing contingent claims in the presence of a higher interest rate for borrowing. In the special case of a European call option in a market with constant coefficients, the endpoints of the arbitrage-free interval are the Black-Scholes prices corresponding to the two different interest rates, and the fair price coincides with that of Barron and Jensen.

In the classical continuous-time financial market model, stock prices have been understood as solutions to linear stochastic differential equations, and an important problem to solve is the problem of hedging options (functions of the stock price values at the expiration date). In this paper we consider the hedging problem not only with a price model that is nonlinear, but also with coefficients of the price equations that can depend on the portfolio strategy and the wealth process of the hedger. In mathematical terminology, the problem translates to solving a forward-backward stochastic differential equation with the forward diffusion part being degenerate. We show that, under reasonable conditions, the four step scheme of Ma, Protter and Yong for solving forward-backward SDE's still works in this case, and we extend the classical results of hedging contingent claims to this new model. Included in the examples is the case of the stock volatility increase caused by overpricing the option, as well as the case of different interest rates for borrowing and lending.

Importance sampling is a Monte Carlo technique where random data are sampled from an alternative "sampling distribution" and an unbiased estimator is obtained by likelihood ratio weighting. Here we consider estimation of large deviations probabilities via importance sampling. Previous works have shown, for certain special cases, that "exponentially twisted" distributions possess a strong asymptotic optimality property as a sampling distribution. The results of this paper unify and generalize the previous special case results. The analysis is presented in an abstract setting, so the results are quite general and directly applicable to a number of large deviations problems. Our main motivation, however, is to attack sample path problems. To illustrate the application to this class of problems, we consider Mogulskii type sample path problems in some detail.

Motivated by applications in texture synthesis, we propose a model selection procedure for Markov random fields based on penalized pseudolikelihood. The procedure is shown to be consistent for choosing the true model, even for Gibbs random fields with phase transitions. As a by-product, rates for the restricted mean-square error and moderate deviation probabilities are derived for the maximum pseudolikelihood estimator. Some simulation results are presented for the selection procedure.

This paper studies several different plans for selecting coordinates for updating via Gibbs sampling. It exploits the inherent features of the Gibbs sampling formulation, most notably its neighborhood structure, to characterize and compare the plans with regard to convergence to equilibrium and variance of the sample mean. Some of the plans rely completely or almost completely on random coordinate selection. Others use completely or almost completely deterministic coordinate selection rules. We show that neighborhood structure induces idempotency for the individual coordinate transition matrices and commutativity among subsets of these matrices. These properties lead to bounds on eigenvalues for the Gibbs sampling transition matrices corresponding to several of the plans. For a frequently encountered neighborhood structure, we give necessary and sufficient conditions for a commonly employed deterministic coordinate selection plan to induce faster convergence to equilibrium than the random coordinate selection plans. When these conditions hold, we also show that this deterministic selection rule achieves the same worst-case bound on the variance of the sample mean as that arising from the random selection rules when the number of coordinates grows without bound. This last result encourages the belief that faster convergence for the deterministic selection rule may also imply a variance of the sample mean no larger than that arising for random selection rules.

We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.

Let ${X_i, 1 \leq i < \infty}$ be i.i.d. with uniform distribution on $[0, 1]^d$ and let $M(X_1, \dots, X_n; \alpha)$ be $\min {\sum_{e \epsilon T'} |e|^{\alpha}; T' \text{a spanning tree on ${X_1, \dots, X_n}$}}$. Then we show that for $\alpha > 0$, $$\frac{M(X_1, \dots, X_n; \alpha) - EM (X_1, \dots, X_n; \alpha)}{n^{(d-2 \alpha)/2d}} \to N(0, \sigma_{\alpha, d}^2)$$ in distribution for some $\sigma_{\alpha, d}^2 > 0$.

We prove that for continuum percolation in $\mathbb{R}^d$, parametrized by the mean number y of points connected to the origin, as $d \to \infty$ with y fixed the distribution of the number of points in the cluster at the origin converges to that of the total number of progeny of a branching process with a Poisson(y) offspring distribution. We also prove that for sufficiently large d the critical points for the existence of infinite occupied and vacant regions are distinct. Our results resolve conjectures made by Avram and Bertsimas in connection with their formula for the growth rate of the length of the Euclidean minimal spanning tree on n independent uniformly distributed points in d dimensions as $n \to \infty$.

Monotonicity properties of certain classes of point processes with respect to the Palm measure are exploited to derive upper and lower bounds on the total variation distance away from Poisson of these processes. The results obtained are applied to new better than used and new worse than used renewal processes and to a Cox process with rates given by a two state Markov chain.

For an arbitrary point of a homogeneous Poisson point process in a d-dimensional Euclidean space, consider the number of Poisson points that have that given point as their rth nearest neighbor $(r = 1, 2, \dots)$. It is shown that as d tends to infinity, these nearest neighbor counts $(r = 1, 2, \dots)$ are iid asymptotically Poisson with mean 1. The proof relies on Rényi's characterization of Poisson processes and a representation in the limit of each nearest neighbor count as a sum of countably many dependent Bernoulli random variables.

A packing of a collection of subintervals of [0, 1] is a pairwise disjoint subcollection of the intervals; its wasted space is the measure of the set of points not covered by the packing.

Consider n random intervals, $I_1, \dots, I_n$, chosen by selecting endpoints independently from the uniform distribution. We strengthen and simplify the results of Coffman, Poonen and Winkler, and we show that, for some universal constant K and for each $t \geq 1$, with probability greater than or equal to $1 - 1/n_t$, there is a packing with wasted space less than or equal to $Kt (\log n)^2 /n$.

We prove that certain (discrete time) probabilistic automata which can be absorbed in a "null state" have a normalized quasi-stationary distribution (when restricted to the states other than the null state). We also show that the conditional distribution of these systems, given that they are not absorbed before time n, converges to an honest probability distribution; this limit distribution is concentrated on the configurations with only finitely many "active or occupied" sites.

A simple example to which our results apply is the discrete time version of the subcritical contact process on $\mathbb{Z}^d$ or oriented percolation on $\mathbb{Z}^d$ (for any $d \geq 1$) as seen from the "leftmost particle." For this and some related models we prove in addition a central limit theorem for $n^{-1/2}$ times the position of the leftmost particle (conditioned on survival until time n).

The basic tool is to prove that our systems are R-positive-recurrent.

Recurrence relations and upper bounds are obtained for power moments of generalized hitting times for semi-Markov processes. General necessary and sufficient conditions for the existence of these moments are also found. Applications to hitting times for semi-Markov dynamical systems of linear type, semi-Markov random walks, diffusion processes and queuing systems are discussed.

This work is concerned with the asymptotic properties of a singular perturbed nonstationary finite state Markov chain. In a recent paper of the authors, it was shown that as the fluctuation rate of the Markov chain goes to $\infty$, the probability distribution of the Markov chain converges to its time-dependent quasi-equilibrium distribution. In addition, asymptotic expansion of the probability distribution was obtained. This paper is a continuation of our effort in this direction. Upon using the asymptotic expansion, a suitably scaled sequence is examined in detail. Asymptotic normality is obtained. It is shown that the accumulated difference between the indicator process and the quasi-equilibrium distribution converges to a Gaussian process with zero mean. An explicit formula for the covariance function of the Gaussian process is obtained, which depends crucially on the asymptotic expansion.

Consider a triangular array of stationary normal random variables ${\xi_{n, i}, i \geq 0, n \geq 1)$ such that ${\xi_{n, i}, i \geq 0}$ is a stationary normal sequence for each $n \geq 1$. Let $\rho_{n, j} = \corr (\xi_{n, i}, \xi_{n, i + j})$. We show that if $(1 - \rho_{n,j}) \log n \to \delta_j \epsilon (0, \infty)$ as $n \to \infty$ for some j, then the locations where the extreme values occur cluster, and if $\rho_{n,j}$ tends to 0 fast enough as $j \to \infty$ for fixed n, then ${\xi_{n, i}, i \geq 0}$ satisfies a certain weak dependence condition. Under the two conditions, it is possible to speak about an index which measures the degree of clustering. In practice, this viewpoint can provide a better approximation of the distributions of the maxima of weakly dependent normal random variables than what is directly guided by the asymptotic theory of Berman.

In this tutorial, statistics on the number of people who tie for first place are considered. It is demonstrated that the so-called Rice's method from the calculus of finite differences is a very convenient tool both to rederive known results as well as to gain new ones with ease.