A continuous-time analog of the Martingale model of forecast evolution

In many practical situations, a manager would like to simulate forecasts for periods whose duration (e.g., week) is not equal to the periods (e.g., month) for which past forecasting data are available. This article addresses this problem by developing a continuous-time analog of the Martingale model of forecast evolution, called the Continuous-Time Martingale Model of Forecast Evolution (CTMMFE). The CTMMFE is used to parameterize the variance–covariance matrix of forecast updates in such a way that the matrix can be scaled for any planning period length. The parameters can then be estimated from past forecasting data corresponding to a specific planning period. Once the parameters are estimated, a variance–covariance matrix can be generated for any planning period length. Numerical experiments are conducted to derive insights into how various characteristics of the variance–covariance matrix (for example, the underlying correlation structure) influence the number of parameters needed as well as the accuracy of the approximation.