Continuous Simulation, Differential Inclusions, Uncertainty, and Traveling in Time

Differential inclusions (DIs) represent an important extension of differential equations. The other term used for DI is differential equations with a set-valued right-hand side. This tool can be used when the model reveals uncertainty that is given in terms of limits or permissible sets rather than random variables. A model of stock exchange dynamics with uncertain information and a model with an ideal predictor are presented to illustrate possible applications. The ideal predictor problem is equivalent to the problem of passing the information from the future to the present, or traveling into the past to use the present information and change the model trajectory. It is shown that the uncertainty over the future can be simulated using differential inclusions. A differential inclusion solver is described.