Explosion

Abstract

Up to this point we have discussed the martingale problem only in connection with bounded coefficients. However it should be evident that many of the more difficult aspects of the martingale problem are really local and do not depend on the global properties of the coefficients. Thus, it should be, and indeed it is, a simple exercise to extend most of the results of Chapters 6, 7 and 9 to martingale problems associated with locally bounded coefficients. The one place at which difficulties can arise is the place where our methods were not local, i.e., in the proof of the existence of solutions to the martingale problem. Although the local existence of solutions is determined completely by the local properties of the coefficients, it is impossible to predict on the basis of local considerations whether a diffusion “ runs out to infinity” in a finite time. Perhaps the best way to see this is to look at the simple ordinary differential equation

$$ \frac{{dx}}{{dy}} = {x^2}(t),{\text{ }}x(0) = 1. $$

((0.1))