A hedger is faced with a commitment in one asset and the opportunity to continuously trade futures contracts on another asset whose returns are correlated with those of the committed asset. Optimal futures trading strategies are presented in closed form for several mean-variance and quadratic objectives.

Multidimensional reflected Brownian motions, also called regulated Brownian motions or simply RBM's, arise as approximate models of queueing networks. Thus the stationary distributions of these diffusion processes are of interest for steady-state analysis of the corresponding queueing systems. This paper considers two-dimensional semimartingale RBM's with rectangular state space, which include the RBM's that serve as approximate models of finite queues in tandem. The stationary distribution of such an RBM is uniquely characterized by a certain basic adjoint relationship, and an algorithm is proposed for numerical solution of that relationship. We cannot offer a general proof of convergence, but the algorithm has been coded and applied to special cases where the stationary distribution can be determined by other means; the computed solutions agree closely with previously known results and convergence is reasonably fast. Our current computer code is specific to two-dimensional rectangles, but the basic logic of the algorithm applies equally well to any semimartingale RBM with bounded polyhedral state space, regardless of dimension. To demonstrate the role of the algorithm in practical performance analysis, we use it to derive numerical performance estimates for a particular example of finite queues in tandem; our numerical estimates of both the throughput loss rate and the average queue lengths are found to agree with simulated values to within about five percent.

We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs. The bounds compare well with exact answers for a variety of simple chains and seem better than bounds derived through Cheeger-like inequalities. They offer improved rates of convergence for the random walk associated to approximate computation of the permanent.

We extend recently developed eigenvalue bounds on mixing rates for reversible Markov chains to nonreversible chains. We then apply our results to show that the $d$-particle simple exclusion process corresponding to clockwise walk on the discrete circle $\mathbf{Z}_p$ is rapidly mixing when $d$ grows with $p$. The dense case $d = p/2$ arises in a Poisson blockers problem in statistical mechanics.

We consider problems involving large or loud values of the shot noise process $X(t) := \sum_{i: \tau_i \leq t} h(t - \tau_i), t \geq 0$, where $h: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is nonincreasing and $(\tau_i, i \geq 0)$ is the sequence of renewal times of a renewal process. Results are obtained by extending the renewal sequence to all $i \in \mathbb{Z}$ and considering the stationary sequence $(\xi_n)$ given by $\xi_n = \sum_{i \leq n} h(\tau_n - \tau_i)$. We show that $\xi_n$ has a thin tail in the sense that under broad circumstances $\operatorname{Pr}\{\xi_n > x + \delta \mid \xi_n > x\} \rightarrow 0$ as $x \rightarrow \infty$, where $\delta > 0$. We also show that $\operatorname{Pr}\{\max(\xi_1, \cdots, \xi_n) \leq u_n\} - (\operatorname{Pr}\{\xi_0 \leq u_n\})^n \rightarrow 0$ for real sequences $(u_n)$ for which $\lim \sup n \operatorname{Pr}\{\xi_0 > u_n\} < \infty$.

Motivated by models of queues with server vacations, we consider a Levy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Levy process so that the overall Levy process with secondary jump input can have a proper limiting distribution. For example, the workload process in an $M/G/1$ queue with a server vacation each time the server finds an empty system is such a Levy process with secondary jump input. We show that a certain functional of a Levy process with secondary jump input is a martingale and we apply this martingale to characterize the steady-state distribution. We establish stochastic decomposition results for the case in which the Levy process has no negative jumps, which extend and unify previous decomposition results for the workload process in the $M/G/1$ queue with server vacations and Brownian motion with secondary jump input. We also apply martingales to provide a new proof of the known simple form of the steady-state distribution of the associated reflected Levy process when the Levy process has no negative jumps (the generalized Pollaczek-Khinchine formula).

Let $X_0$ be a nonnegative integer-valued random variable and let an independent copy of $X_0$ be assigned to each leaf of a binary tree of depth $k$. If $X_0$ and $X'_0$ are adjacent leaves, let $X_1 = (X_0 - 1)^+ + (X'_0 - 1)^+$ be assigned to the parent node. In general, if $X_j$ and $X'_j$ are assigned to adjacent nodes at level $j = 0, \cdots, k - 1$, then $X_j$ and $X'_j$ are, in turn, independent and the value assigned to their parent node is then $X_{j+1} = (X_j - 1)^+ + (X'_j - 1)^+$. We ask what is the behavior of $X_k$ as $k \rightarrow \infty$. We give sufficient conditions for $X_k \rightarrow \infty$ and for $X_k \rightarrow 0$ and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part).

For the solutions of certain random equations, or equivalently the stationary solutions of certain random recurrences, the distribution tails are evaluated by renewal-theoretic methods. Six such equations, including one arising in queueing theory, are studied in detail. Implications in extreme-value theory are discussed by way of an illustration from economics.

Systems defined by $dx(t) = a\lbrack x(t), t \rbrack dt + B\lbrack x(t), t \rbrack u(t) dt + N^{1/2}\lbrack x(t), t \rbrack dW(t)$, where $x(t)$ is the state variable, $u(t)$ is the control variable, $a$ is a vector function, $B$ and $N$ are matrices and $W(t)$ is a Brownian motion process, are considered. The aim is to minimize the expected value of a cost function with quadratic control costs on the way and terminal cost function $K(T)$, where $T = \inf\{s: x(s) \in D \mid x(t) = x\}, D$ being a given region in $\mathbb{R}^n$. The function $K$ is taken to be 0 if $T \geq (\leq) \tau$, where $\tau$ is a positive constant and $+\infty$ if $T < (>) \tau$ when the aim is to force $x(t)$ to stay in (resp., to leave) the region $C$, the complement of $D$. A particular one-dimensional problem is solved explicitly and a risk-sensitive version of the general problem is also considered.