Information relaxations and duality in stochastic dynamic programs

We describe a general technique for determining upper bounds on maximal values (or lower bounds on minimal costs) in stochastic dynamic programs. In this approach, we relax the nonanticipativity constraints that require decisions to depend only on the information available at the time a decision is made and impose a "penalty" that punishes violations of nonanticipativity. In applications, the hope is that this relaxed version of the problem will be simpler to solve than the original dynamic program. The upper bounds provided by this dual approach complement lower bounds on values that may be found by simulating with heuristic policies. We describe the theory underlying this dual approach and establish weak duality, strong duality, and complementary slackness results that are analogous to the duality results of linear programming. We also study properties of good penalties. Finally, we demonstrate the use of this dual approach in an adaptive inventory control problem with an unknown and changing demand distribution and in valuing options with stochastic volatilities and interest rates. These are complex problems of significant practical interest that are quite difficult to solve to optimality. In these examples, our dual approach requires relatively little additional computation and leads to tight bounds on the optimal values. © 2010 INFORMS.

Equity trading volume and volatility: Latent information arrivals and common long-run dependencies

This article examines the behavior of equity trading volume and volatility for the individual firms composing the Standard & Poor's 100 composite index. Using multivariate spectral methods, we find that fractionally integrated processes best describe the long-run temporal dependencies in both series. Consistent with a stylized mixture-of-distributions hypothesis model in which the aggregate "news"-arrival process possesses long-memory characteristics, the long-run hyperbolic decay rates appear to be common across each volume-volatility pair.

Eliciting and Aggregating Forecasts When Information is Shared

Using the wisdom of crowds---combining many individual forecasts to obtain an aggregate estimate---can be an effective technique for improving forecast accuracy. When individual forecasts are drawn from independent and identical information sources, a simple average provides the optimal crowd forecast. However, correlated forecast errors greatly limit the ability of the wisdom of crowds to recover the truth. In practice, this dependence often emerges because information is shared: forecasters may to a large extent draw on the same data when formulating their responses.

To address this problem, I propose an elicitation procedure in which each respondent is asked to provide both their own best forecast and a guess of the average forecast that will be given by all other respondents. I study optimal responses in a stylized information setting and develop an aggregation method, called pivoting, which separates individual forecasts into shared and private information and then recombines these results in the optimal manner. I develop a tailored pivoting procedure for each of three information models, and introduce a simple and robust variant that outperforms the simple average across a variety of settings.

In three experiments, I investigate the method and the accuracy of the crowd forecasts. In the first study, I vary the shared and private information in a controlled environment, while the latter two studies examine forecasts in real-world contexts. Overall, the data suggest that a simple minimal pivoting procedure provides an effective aggregation technique that can significantly outperform the crowd average.