does not hit 1, almost surely), and is a local martingale, since the function is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as Nevertheless, the expectation of this process is non-constant; moreover, as which can be deduced from the fact that the mean value of over the circle tends to infinity as . (In fact, it is equal to for r ≥ 1 but to 0 for r ≤ 1). Martingales via local martingales Let be a local martingale. In order to prove that it is a martingale it is sufficient to prove that in L1 (as ) for every t, that is, here is the stopped process. The given relation implies that almost surely. The dominated convergence theorem ensures the convergence in L1 provided that for every t. Thus, Condition (*) is sufficient for a local martingale being a martingale. A stronger condition for every t is also sufficient. Caution. The weaker condition for every t is not sufficient. Moreover, the condition is still not sufficient; for a counterexample see Example 3 above. A special case: where is the Wiener process, and is twice continuously differentiable. The process is a local martingale if and only if f satisfies the PDE However, this PDE itself does not ensure that is a martingale. In order to apply (**) the following condition on f is sufficient: for every and t there exists such that for all and Technical details References Martingale theory