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Given a real valued and time-inhomogeneous martingale diffusion X, weinvestigate the properties of functions defined by the conditional expectationf(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitzcontinuous then f(t,x) will also be monotonic or Lipschitz continuous in x. Ifg is convex then f(t,x) will be convex in x and decreasing in t. We also definethe marginal support of a process and show that it almost surely contains thepaths of the process. Although f need not be jointly continuous, we show thatit will be continuous on the marginal support of X. We prove these results fora generalization of diffusion processes that we call `almost-continuousdiffusions', and includes all continuous and strong Markov processes.

Properties of Expectations of Functions of Martingale Diffusions