Consider a tree network $T$, where each edge acts as an independent copy of a given channel $M$, and information is propagated from the root. For which $T$ and $M$ does the configuration obtained at level $n$ of $T$ typically contain significant information on the root variable? This problem arose independently in biology, information theory and statistical physics.

For all $b$, we construct a channel for which the variable at the root of the break $b$-ary tree is independent of the configuration at the second level of that tree, yet for sufficiently large $B>b$, the mutual information between the configuration at level $n$ of the $B$-ary tree and the root variable is bounded away from zero for all $n$. This construction is related to Reed--Solomon codes.

We improve the upper bounds on information flow for asymmetric binary channels (which correspond to the Ising model with an external field) and for symmetric $q$-ary channels (which correspond to Potts models).

Let $\lam_2(M)$ denote the second largest eigenvalue of $M$, in absolute value. A CLT of Kesten and Stigum implies that if $b |\lam_2(M)|^2 >1$, then the census of the variables at any level of the $b$-ary tree, contains significant information on the root variable. We establish a converse: If $b |\lam_2(M)|^2 < 1$, then the census of the variables at level $n$ of the $b$-ary tree is asymptotically independent of the root variable. This contrasts with examples where $b |\lam_2(M)|^2 <1$, yet the configuration at level $n$ is not asymptotically independent of the root variable.

We present the stochastic approach to nonlinear kinetic equations (without gradient terms) in a unifying general framework, which covers many interactions important in applications, such as coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. We provide conditions for the existence of corresponding stochastic particle systems in the sense of regularity (nonexplosion) of a jump process with unbounded intensity. Using an appropriate space of measure-valued functions, we prove relative compactness of the sequence of processes and characterize the weak limits in terms of solutions to the nonlinear equation. As a particular application, we derive existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with inelastic collisions.

We use a notion of stochastic time, here called volatility time, to show convexity of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the volatility. The volatility time is obtained as the almost surely unique stopping time solution to a random ordinary differential equation related to volatility. This enables us to write price processes, or processes modeled by local martingales, as Brownian motions with respect to volatility time. The results are shown under very weak assumptions and are of independent interest in the study of stochastic differential equations. Options on several underlying assets are also studied and we prove that if the volatility matrix is independent of time, then the option prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets and the option prices do not necessarily decay with time, if a time-dependent volatility is allowed.

Kunitomo and Takahashi (1995, 2001) have proposed a new methodology, called small disturbance asymptotics, for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. It can be applicable to a wide range of valuation problems, including complicated contingent claims associated with the Black--Scholes model and the term structure model of interest rates in the Heath--Jarrow--Morton framework. Our approach can be rigorously justified by an infinite-dimensional analysis called the Watanabe--Yoshida theory on the Malliavin calculus recently developed in stochastic analysis.

This paper examines the valuation of call options on the minimum of two dividend-paying assets. We show that the optimal exercise boundary consists of three components, two continuous curves and one component along the diagonal with empty interior. The option price is shown to satisfy the early exercise premium representation in which the gains from exercise involve the local time of the minimum of the two underlying asset prices. A system of recursive integral equations for the exercise boundary components is derived. Using a class of simple stopping times we also construct lower and upper bounds for the American call min-option price: these are easy to compute and can be employed to design efficient approximations of the contract value.

We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuous-state branching processes with immigration and Ornstein--Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.

We study the infinite time ruin probability for an insurance company in the classical Cramér--Lundberg model with finite exponential moments. The additional nonclassical feature is that the company is also allowed to invest in some stock market, modeled by geometric Brownian motion. We obtain an exact analogue of the classical estimate for the ruin probability without investment, that is, an exponential inequality. The exponent is larger than the one obtained without investment, the classical Lundberg adjustment coefficient, and thus one gets a sharper bound on the ruin probability.

A surprising result is that the trading strategy yielding the optimal asymptotic decay of the ruin probability simply consists in holding a fixed quantity (which can be explicitly calculated) in the risky asset, independent of the current reserve. This result is in apparent contradiction to the common believe that "rich" companies should invest more in risky assets than "poor" ones. The reason for this seemingly paradoxical result is that the minimization of the ruin probability is an extremely conservative optimization criterion, especially for "rich" companies.

In this article it is shown that one is able to evaluate the price of perpetual calls, puts, Russian and integral options directly as the Laplace transform of a stopping time of an appropriate diffusion using standard fluctuation theory. This approach is offered in contrast to the approach of optimal stopping through free boundary problems. Following ideas of Carr [Rev. Fin. Studies 11 (1998) 597--626], we discuss the Canadization of these options as a method of approximation to their finite time counterparts. Fluctuation theory is again used in this case.

The Leland strategy for an approximate hedging of the call option under transactions costs is studied. The rate of convergence in the Kabanov--Safarian theorem for the Leland strategy is found. The limit theorem for the hedging portfolio is proved.

This paper describes a general type of stochastic system model that involves three basic elements: activities, resources, and stocks of material. A system manager chooses activity levels dynamically based on state observations, consuming some materials as inputs and producing other materials as outputs, subject to resource capacity constraints. A generalized notion of heavy traffic is described, in which exogenous input and output rates are approximately balanced with nominal activity rates derived from a static planning problem. A Brownian network model is then proposed as a formal approximation in the heavy traffic parameter regime. The current formulation is novel, relative to models analyzed in previous work, in that its definition of heavy traffic takes explicit account of the system manager's economic objective.

We consider a multiclass queueing network with $N$ customer classes, each having an arbitrary fixed route through the network. (Thus, the network is not necessarily feedforward.) We show that the largest weighted delay first (LWDF) discipline is an optimal scheduling discipline in the network in the following sense. Let $w_i$ be the (random) instantaneous largest end-to-end delay of a class $i$ customer in the network in stationary regime. For any set of positive constants $\alpha_1, \ldots, \alpha_N$, the LWDF discipline associated with this set maximizes (among all disciplines) the quantity \[ (1)\hspace*{15pt}\min_{i=1,\ldots,N} \left[ \alpha_i \lim_{n\to\infty} \frac{-1}{n} \log P \lb w_i > n \rb \right] = \lim_{n\to\infty} \frac{-1}{n} \log P \lb r > n \rb,\hspace*{15pt} \] where $r \doteq \max_{i} w_i/\alpha_i$ is the maximal weighted delay in the network. [This result is a generalization of the single-server result proved by A. L. Stolyar and K. Ramanan in Ann. Appl. Probab. 11 (2001) 1--48.]

As the key element of the proof, we establish the following critical node property: In a LWDF network, there exists a most likely path to build large $r$, which is a most likely path to do so in one of the network nodes in isolation. Such a most likely path has a very simple structure: its parameters [and the optimal value of (1)] can be computed by solving a finite-dimensional optimization problem for each network node.

By using some quasi-renewal-like equations and functional differential equations, we explicitly compute the Laplace transforms of some random variables introduced by Cowan and Chiu in modeling the mechanism of replication of a DNA molecule [J. Appl. Probab. (1994) 31 301--308]. These Laplace transforms are expressed by means of infinite products arising in the theory of partitions.