**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 *L*1 (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 *L*1 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

Read more about this topic: Local Martingale

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