Editing Wiener process (section)

=== Time change ===
Every continuous martingale (starting at the origin) is a time changed Wiener process.

'''Example:'''  2''W''<sub>''t''</sub> = ''V''(4''t'') where ''V'' is another Wiener process (different from ''W'' but distributed like ''W'').

'''Example.''' <math> W_t^2 - t = V_{A(t)} </math> where <math> A(t) = 4 \int_0^t W_s^2 \, \mathrm{d} s </math> and ''V'' is another Wiener process.

In general, if ''M'' is a continuous martingale then <math> M_t - M_0 = V_{A(t)} </math> where ''A''(''t'') is the [[quadratic variation]] of ''M'' on [0, ''t''], and ''V'' is a Wiener process.

'''Corollary.''' (See also [[Doob's martingale convergence theorems]]) Let ''M<sub>t</sub>'' be a continuous martingale, and
<math display="block">M^-_\infty = \liminf_{t\to\infty} M_t,</math>
<math display="block">M^+_\infty = \limsup_{t\to\infty} M_t. </math>

Then only the following two cases are possible:
<math display="block"> -\infty < M^-_\infty = M^+_\infty < +\infty,</math>
<math display="block">-\infty = M^-_\infty < M^+_\infty = +\infty; </math>
other cases (such as <math> M^-_\infty = M^+_\infty = +\infty, </math> &nbsp; <math> M^-_\infty < M^+_\infty < +\infty </math> etc.) are of probability 0.

Especially, a nonnegative continuous martingale has a finite limit (as ''t'' → ∞) almost surely.

All stated (in this subsection) for martingales holds also for [[local martingale]]s.