This paper summarizes the essential results on the pricing of the American option.

Let $S$ be a compact metric space, let $\theta > 0$, and let $P(x,dy)$ be a one-step Feller transition function on $S \times \mathscr{B}(S)$ corresponding to a weakly ergodic Markov chain in $S$ with unique stationary distribution $\nu_0$. The neutral diffusion model, or Fleming-Viot process, with type space $S$, mutation intensity $\frac{1}{2}\theta$ and mutation transition function $P(x,dy)$, assumes values in $\mathscr{P}(S)$, the set of Borel probability measures on $S$ with the topology of weak convergence, and is known to be weakly ergodic and have a unique stationary distribution $\Pi \in \mathscr{P}(\mathscr{P}(S))$. Define the Markov chain $\{X(\tau), \tau \in \mathbf{Z}_+\}$ in $S^2 \cup S^3 \cup \cdots$ as follows. Let $X(0) = (\xi,\xi) \in S^2$, where $\xi$ is an $S$-valued random variable with distribution $\nu_0$. From state $(x_1, \ldots, x_n) \in S^n$, where $n \geq 2$, one of two types of transitions occurs. With probability $\theta/(n(n - 1 + \theta))$ a transition to state $(x_1,\ldots,x_{i - 1},\xi_i,x_{i + 1}, \ldots, x_n) \in S^n$ occurs $(1 \leq i \leq n)$, where $\xi_i$ is distributed according to $P(x_i,dy)$. With probability $(n - 1)/((n + 1)n(n - 1 + \theta))$ a transition to state $(x_1, \ldots ,x_{j - 1},x_i,x_j, \ldots, x_n) \in S^{n + 1}$ occurs $(1 \leq i \leq n, 1 \leq j \leq n + 1)$. Letting $\tau_n$ denote the hitting time of $S^n$, we show that the empirical measure determined by the $n$ coordinates of $X(\tau_{n + 1} - 1)$ converges almost surely as $n \rightarrow \infty$ to a $\mathscr{P}(S)$-valued random variable with distribution $\Pi$.

We provide sufficient conditions for the existence of a solution to a consumption and portfolio problem in continuous time under uncertainty with an infinite horizon. When the price processes for securities are diffusion processes, optimal policies can be computed by solving a linear partial differential equation. We also provide conditions under which the solution to an infinite horizon problem is the limit of the solutions to finite horizon problems when the horizon increases to infinity.

This paper is concerned with a class of multidimensional diffusion processes, variously known as reflected Brownian motions, regulated Brownian motions, or just RBM's, that arise as approximate models of queueing networks. We develop an algorithm for numerical analysis of a semimartingale RBM with state space $S = \mathbb{R}^d_+$ (the nonnegative orthant of $d$-dimensional Euclidean space). This algorithm lies at the heart of the QNET method for approximate two-moment analysis of open queueing networks.

A continuous-time, consumption-investment problem on a finite horizon is considered for an agent seeking to maximize expected utility from consumption plus expected utility from terminal wealth. The agent is prohibited from selling stocks short, so the usual martingale methods for solving this problem do not directly apply. A dual problem is posed and solved, and the solution to the dual problem provides information about the existence and nature of the solution to the original problem.

Given $n$ uniformly and independently distributed points in the $d$-dimensional cube of unit volume, it is well established that the length of the minimum spanning tree on these $n$ points is asymptotic to $\beta_{\mathrm{MST}}(d)n^{(d - 1)/d}$, where the constant $\beta_{\mathrm{MST}}(d)$ depends only on the dimension $d$. It has been a major open problem to determine the constant $\beta_{\mathrm{MST}}(d)$. In this paper we obtain an exact expression for the constant $\beta_{\mathrm{MST}}(d)$ on a torus as a series expansion. Truncating the expansion after a finite number of terms yields a sequence of lower bounds; the first five terms give a lower bound which is already very close to the empirically estimated value of the constant. Our proof technique unifies the derivation for the MST asymptotic behavior for the Euclidean and the independent model.

A space-time jump Markov process, modeling a chemical reaction with diffusion, is compared in the supremum norm to the usual model, the solution to a partial differential equation. Conditions are given which imply the deviation converges in probability to 0 uniformly on bounded time intervals. Estimates reflecting underlying large deviation behavior are obtained.

We study parameter estimation for Markov random fields (MRFs) over $Z^d, d \geq 1$, from incomplete (degraded) data. The MRFs are parameterized by points in a set $\Theta \subseteq \mathbb{R}^m, m \geq 1$. The interactions are translation invariant but not necessarily of finite range, and the single-pixel random variables take values in a compact space. The observed (degraded) process $y$ takes values in a Polish space, and it is related to the unobserved MRF $x$ via a conditional probability $P^{y \mid x}$. Under natural assumptions on $P^{y \mid x}$, we show that the ML estimations are strongly consistent irrespective of phase transitions, ergodicity or stationarity, provided that $\Theta$ is compact. The same result holds for noncompact $\Theta$ under an extra assumption on the pressure of the MRFs.

We consider a hedger with a mean-variance objective who faces a random loss at a fixed time. The size of this loss depends quite generally on two correlated asset prices, while only one of them is available for hedging purposes. We present a simple solution of this hedging problem by introducing the intrinsic value process of a contingent claim.

We establish general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters. The asymptotic validity occurs as the prescribed volume of the confidence set approaches 0. There are two requirements: a functional central limit theorem for the estimation process and strong consistency (with-probability-1 convergence) for the variance or "scaling matrix" estimator. Applications are given for: sample means of i.i.d. random variables and random vectors, nonlinear functions of such sample means, jackknifing, Kiefer-Wolfowitz and Robbins-Monro stochastic approximation and both regenerative and nonregenerative steady-state simulation.

We derive large deviation approximations to boundary crossing probabilities for a class of point processes which can be approximated locally as Poisson processes. In the special case of empirical processes, we are able to obtain second order correction terms. The methods are applied to Kolmogorov-Smirnov testing, where we are able to obtain accurate approximations to the significance level when the null hypothesis is an exponential family with unknown nuisance parameters.

The $3x + 1$ problem concerns the behavior under iteration of the function $T: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ defined by $T(n) = n/2$ if $n$ is even and $T(n) = (3n + 1)/2$ if $n$ is odd. The $3x + 1$ conjecture asserts that for each $n \geq 1$ some $k$ exists with $T^{(k)}(n) = 1$; let $\sigma_\infty(n)$ equal the minimal such $k$ if one exists and $+\infty$ otherwise. The behavior of $\sigma_\infty(n)$ is irregular and seems to defy simple description. This paper describes two kinds of stochastic models that mimic some of its features. The first is a random walk that imitates the behavior of $T (\operatorname{mod}2^j)$; the second is a family of branching random walks that imitate the behavior of $T^{-1} (\operatorname{mod}3^j)$. For these models we prove analogues of the conjecture that $\lim \sup_{n \rightarrow \infty}(\sigma_\infty(n)/\log(n)) = \gamma$ for a finite constant $\gamma$. Both models produce the same constant $\gamma_0 \doteq 41.677647$. Predictions of the stochastic models agree with empirical data for the $3x + 1$ problem up to $10^{11}$. The paper also studies how many $n$ have $\sigma_\infty(n) = k$ as $k \rightarrow \infty$ and estimates how fast $t(n) = \max(T^{(k)}(n): k \geq 0)$ grows as $n \rightarrow \infty$.