By a filtered Monte Carlo estimator we mean one whose constituent parts—summands or integral increments—are conditioned on an increasing family of σ-fields. Unbiased estimators of this type are suggested by compensator identities. Replacing a point-process integrator with its intensity gives rise to one class of examples; exploiting Levy's formula gives rise to another.
Money managers select weights of managed portfolios to enhance their reputation in the spot market for their services, inevitably using their actions to signal their quality. We develop a two-asset signalling model of money managers. A unique screening equilibrium and (under certain parameter configurations) a host of pooling equilibria survive the Cho-Kreps Intuitive Criterion. In all the equilibria managers behave more aggressively than they would in the absence of the signalling motive, exaggerating their position in the risky asset.
Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d.
Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simultation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias.
If the score in a squash game is tied late in the game, one player has a choice of how many additional points (from a prespecified set of possibilities) are to be played to determine the winner. This paper constructs a Markov chain model of the situation and solves for the optimal strategy. Expressions for the optimal strategy are obtained with a symbolic algebra computer package. Results are given for both international and American scoring systems. The model and analysis are very suitable for educational purposes.
This essay reviews the literature on the role of the financial factors in the Depression, and draws some lessons that have more general relevance for the study of the Depression and for macroeconomics. I argue that much of the recent progress that has been made in understanding some of the most important and puzzling aspects of financial-real links in the Depression followed a paradigm shift in economics.
We investigate the stability of waiting-time derivatives when inputs to a queueing system-service times and interarrival times-depend on a parameter. We give conditions under which the sequence of waiting-time derivatives admits a stationary distribution, and under which the derivatives converge to the stationary regime from all initial conditions. Further hypotheses ensure that the expectation of a stationary waiting-time derivative is, in fact, the derivative of the expected stationary waiting time.
Common random numbers (CRN) is a widely-used technique for reducing variance in comparing stochastic systems through simulation. Its popularity derives from its intuitive appeal and ease of implementation. However, though CRN has been observed to work well with a broad range of models, the class of systems for which it is provably advantageous has remained rather limited.
We derive conditions under which the increments of a vector process are associated—i.e., under which all pairs of increasing functions of the increments are positively correlated. The process itself is associated if it is generated by a family of associated and monotone kernels. We show that the increments are associated if the kernels are associated and, in a suitable sense, convex. In the Markov case, we note a connection between associated increments and temporal stochastic convexity.