Local Martingale - Martingales Via Local Martingales

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