Properties of Expectations of Functions of Martingale Diffusions
نویسندگان
چکیده
Given a real valued and time-inhomogeneous martingale diffusion X , we investigate the properties of functions defined by the conditional expectation f(t,Xt) = E [g(XT )|Ft]. We show that whenever g is monotonic or Lipschitz continuous then f(t, x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t, x) will be convex in x and decreasing in t. We also define the marginal support of a process and show that it almost surely contains the paths of the process. Although f need not be jointly continuous, we show that it will be continuous on the marginal support ofX . We prove these results for a generalization of diffusion processes that we call ‘almost-continuous diffusions’, and includes all continuous and strong Markov processes.
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