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
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