We consider a positive propagator that is driven by time-inhomogeneous Markov processes. We multiply the propagator with a time-dependent, decreasing positive weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. Such supermartingales are suitable for the modelling of the pricing kernel in the case where it is assumed to be given by a function of time and Markov processes. This situation is encountered for example, if we assume that the pricing kernel is sensitive to partial information about economic factors, and the partial information is modelled by use of time-inhomogeneous Markov processes. We show how closed-form expressions for bond prices along with the associated interest-rate and market price of risk models can be obtained, and indicate the way towards the pricing of fixed-income derivatives within this framework. (In collaboration with Jiro Akahori, Ritsumeikan University)