Properties of Expectations of Functions of Martingale Diffusions
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Publication:6207981
arXiv0801.0330MaRDI QIDQ6207981
Publication date: 1 January 2008
Abstract: Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. 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 of X. We prove these results for a generalization of diffusion processes that we call `almost-continuous diffusions', and includes all continuous and strong Markov processes.
Continuous-time Markov processes on general state spaces (60J25) Martingales with continuous parameter (60G44) Diffusion processes (60J60)
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