Optimal heating or cooling of a building space under continuous time uncertainty

A single PDE models the economics and dynamics, in continuous time, of how a building space responds to a stochastic external temperature and to a heating or cooling system. A quadratic function models user discomfort (during intermittent occupation) and we assume a daily cycle of electricity prices (stepwise or continuous). We can rapidly compute the time-varying optimal temperature control rule (the precision of control varies optimally with variations in the cost of control). We can also rapidly compute a purely physical performance parameter (mean or variance) for any variable, over any region of the problem space, plus the time to first exit from that region.