We propose a new put option where the option buyer receives the maximum price (discounted) that the option has ever traded at during the time period (which may be indefinitely long) between the purchase time and the exercise time, so that the buyer need look at the fluctuations only occasionally and enjoys having little or no regret that he did not exercise the option at an earlier time (except for the discounting). We give an exact simple formula for the optimal expected present value (fair price) that can be derived from the option and the (unique) optimal exercise strategy that achieves the optimum value under the assumption that the asset fluctuations follow the Black-Scholes exponential Brownian motion model, which is widely accepted. It is important to note that the discounting is necessary: If it is omitted or even if it is less than the Black-Scholes drift, then the value to the buyer under optimum performance is infinite. We also solve the same problem under a different model: the original Bachelier linear Brownian market with linear discounting. This model is no longer accepted, but of course the mathematics is consistent. To our knowledge no such regretless option is currently traded in any existing market despite its evident appeal. We call it the Russian option, partly to distinguish it from the American and European options, where the term of the option is prescribed in advance and where no exact formula for the value has been given.

Let $X = \{X_t,t \geq 0\}$ be the price process for a stock, with $X_0 = x > 0$. Given a constant $s \geq x$, let $S_t = \max\{s,\sup_{0\leq u \leq t} X_u\}$. Following the terminology of Shepp and Shiryaev, we consider a "Russian option," which pays $S_\tau$ dollars to its owner at whatever stopping time $\tau \in \lbrack 0,\infty)$ the owner may select. As in the option pricing theory of Black and Scholes, we assume a frictionless market model in which the stock price process $X$ is a geometric Brownian motion and investors can either borrow or lend at a known riskless interest rate $r > 0$. The stock pays dividends continuously at the rate $\delta X_t$, where $\delta \geq 0$. Building on the optimal stopping analysis of Shepp and Shiryaev, we use arbitrage arguments to derive a rational economic value for the Russian option. That value is finite when the dividend payout rate $\delta$ is strictly positive, but is infinite when $\delta = 0$. Finally, the analysis is extended to perpetual lookback options. The problems discussed here are rather exotic, involving infinite horizons, discretionary times of exercise and path-dependent payouts. They are also perfectly concrete, which allows an explicit, constructive treatment. Thus, although no new theory is developed, the paper may serve as a useful tutorial on option pricing concepts.

We employ a stochastic control approach to study the question of hedging contingent claims by portfolios constrained to take values in a given closed, convex subset of $\mathscr{R}^d$. In the framework of our earlier work for utility maximization with constrained portfolios, we extend results of El Karoui and Quenez on incomplete markets and treat the case of different interest rates for borrowing and lending.

A group of stations in parallel is considered, where the input processes to the various stations are stochastically dependent nondecreasing Levy processes and the release rates are deterministic linear flows. It is shown that the model of a tandem fluid network, with dependent or independent Levy inputs to the stations, can be seen as a special case of this construction. In addition to other natural applications, the joint stochastic structure of the workload processes of the different classes in a preemptive resume priority M/G/1 queue can be viewed as a special case of the model considered. Using martingale and regenerative arguments, certain steady state characteristics are studied.

We introduce geometric comparison inequalities that give bounds on the eigenvalues of a reversible Markov chain in terms of the eigenvalues of a second chain. The bounds are applied to get sharp results for the exclusion process.

Suppose that the scores of $n$ players are independent integer-valued random variables with probabilities $p_j$. We study the probability $P(T_n)$ that there is a tie for the highest score. The asymptotic behavior of this probability is surprising. Depending on the limit of $p_{j+1}/p_j$, we find different limits of different subsequences $P(T_{n(m)})$. These limits are evaluated for several families of discrete distributions.

Random colorings (independent or dependent) of $\mathbb{Z}^d$ give rise to dependent first-passage percolation in which the passage time along a path is the number of color changes. Under certain conditions, we prove strict positivity of the time constant (and a corresponding asymptotic shape result) by means of a theorem of Cox, Gandolfi, Griffin and Kesten about "greedy" lattice animals. Of particular interest are i.i.d. colorings and the $d = 2$ Ising model. We also apply the greedy lattice animal theorem to prove a result on the omnipresence of the infinite cluster in high density independent bond percolation.

We consider a continuous-time risk process $\{Y_a(t); t \geq 0\}$ defined for a stationary marked point process $\{(T_n,X_n)\}$, where $Y_a(0) = a$ and $Y_a(t)$ increases linearly with a rate $c$ and has a downward jump at time $T_n$ with jump size $X_n$ for $n \in \{1,2,\ldots\}$. For $a = 0$, we prove that, under a balance condition, the descending ladder height distribution of $\{Y_0(t)\}$ has the same form as the case where $\{(T_n,X_n)\}$ is a compound Poisson process. This generalizes the recent result of Frenz and Schmidt, in which the independence of $\{T_n\}$ and $\{X_n\}$ is assumed. In our proof, a differential equation is derived concerning the deficit $Z_a$ at the ruin time of the risk process $\{Y_a(t)\}$ for an arbitrary $a \geq 0$. It is shown that this differential equation is also useful for proving a continuity property of ladder height distributions.

This paper shows the existence and uniqueness of the solution of a backward stochastic differential equation inspired from a model for stochastic differential utility in finance theory. We show our results assuming, when possible, no more than the integrability of the terms involved in the equation. We also show the existence and uniqueness of the solution of a backward-forward stochastic differential equation, where the solution depends explicitly on both the past and the future of its own trajectory, under a more restrictive hypothesis on the Lipschitz constant.

A classical paper by Steele establishes a limit theorem for a wide class of random processes that arise in problems of geometric probability. We propose a different (and arguably more general) set of conditions under which complete convergence holds. As an application of our framework, we prove complete convergence of $M(X_1, \ldots, X_n)/\sqrt n$, where $M(X_1, \ldots, X_n)$ denotes the shortest sum of the lengths of $\lfloor n/2\rfloor$ segments that match $\lfloor n/2\rfloor$ disjoint pairs of points among $X_1, \ldots, X_n$, where the random variables $X_1, \ldots, X_n, \ldots$ are independent and uniformly distributed in the unit square.

Given $N$ points on a circle, a selection-replacement operation removes one of the points and replaces it by another. To select the removed point, an extra point $P$, uniformly distributed, arrives at random and starts moving counterclockwise around the circle; $P$ removes the first point it encounters. A new random point, uniformly distributed, then replaces the removed point. The quantity of interest is $d = d(N)$, the distance that the searching point $P$ must travel to select a point. After many repeated selection-replacements, the joint probability distribution of the $N$ points tends to a stationary limit. We examine the mean of $d$ in this limit. Exact means are found for $N \leq 3$. For large $N$, the mean grows like $(\log^{3/2} N)/N$. These means are larger than the means $1/(N + 1)$ that would be obtained with $N$ independent uniformly distributed points because the selection mechanism tends to cluster the $N$ points into clumps. In a computer application, the circle represents a track on a disk memory, $P$ is a read-write head, the $N$ points mark the beginnings of $N$ files and $d$ determines the time wasted as the head moves from the end of the last file processed to the beginning of the next. $N$ is a parameter of the service rule (the next service goes to one of the $N$ customers waiting the longest).

We consider a version of the data augmentation algorithm of Tanner and Wong, which is a special case of the Gibbs sampler. Using ideas from Harris recurrence, we derive quantitative, a priori bounds on the number of iterations required to achieve convergence. Our analysis involves relating the Markov chain to an associated dynamical system.

For a class of birth and death processes under a heavy traffic condition, the asymptotic behavior of functionals of their jumps is investigated. It is shown that, under suitable normalization, the functionals converge in law to a process that is the sum of a Brownian motion and a constant times the local time of a reflecting Brownian motion at zero. Some applications to queueing processes are presented.

Two kinds of stationary point process are considered. One is generated by the sequence of death times in a simple immigration, birth and death process; the other is the Cox process with intensity given by the square of the radial Ornstein-Uhlenbeck process. By comparison of the coincidence densities, we show that the two classes of processes are equivalent. An explicit expression is given for the coincidence density of arbitrary order.

This paper studies the asymptotic mean squared error for predicting the integral of a weakly stationary spatial process over a unit cube based on a centered systematic sample. For processes whose spectral density decays sufficiently slowly at infinity, the asymptotic mean squared error takes a form similar to that obtained by letting the cube increase in size with the number of observations. However, if the spectral density decays faster than a certain critical rate, then the asymptotic mean squared error takes on a completely different form. By adjusting the weights given to observations near an edge of the cube, it is possible to obtain asymptotic results for the fixed cube that again resemble those for the increasing cube.

We study light traffic approximations for queues in series with renewal arrivals and i.i.d. service time vectors. Formulae for limits of functions of the waiting time at different stations based on single customer effect are obtained for two approaches: dilation and thinning of the arrival process. Interdeparture times from a station possess a one-dependence property in light traffic. This paper complements previous studies of Daley and Rolski and also Asmussen's approach to light traffic limits applied to the cases considered.

Let $\pi(x)$ be a given probability density proportional to $\exp(-U(x))$ in a high-dimensional Euclidean space $\mathbb{R}^m$. The diffusion $dX(t) = -\nabla U(X(t))dt + \sqrt 2 dW(t)$ is often used to sample from $\pi$. Instead of $-\nabla U(x)$, we consider diffusions with smooth drift $b(x)$ and having equilibrium $\pi(x)$. First we study some general properties and then concentrate on the Gaussian case, namely, $-\nabla U(x) = Dx$ with a strictly negative-definite real matrix $D$ and $b(x) = Bx$ with a stability matrix $B$; that is, the real parts of the eigenvalues of $B$ are strictly negative. Using the rate of convergence of the covariance of $X(t)$ [or together with $EX(t)$] as the criterion, we prove that, among all such $b(x)$, the drift $Dx$ is the worst choice and that improvement can be made if and only if the eigenvalues of $D$ are not identical. In fact, the convergence rate of the covariance is $\exp(2\lambda_M(B)t)$, where $\lambda_M(B)$ is the maximum of the real parts of the eigenvalues of $B$ and the infimum of $\lambda_M(B)$ over all such $B$ is $1/m \operatorname{tr} D$. If, for example, a "circulant" drift $\bigg(\frac{\partial U}{\partial x_m} - \frac{\partial U}{\partial x_2},\frac{\partial U}{\partial x_1} - \frac{\partial U}{\partial x_3}, \cdots, \frac{\partial U}{\partial x_{m-1}} - \frac{\partial U}{\partial x_1}\bigg)$ is added to $Dx$, then for essentially all $D$, the diffusion with this modified drift has a better convergence rate.

This existence of an asymptotically efficient importance sampling distribution for estimating small probabilities of statistics of Gibbs random fields is proved, together with its uniqueness, in an appropriate sense. This distribution is also a Gibbs random field associated with an interaction potential that is explicitly given. The particular case of Markov chains is treated. The practical relevance of the results in applications is discussed.