Editing Wiener process (section)

=== A class of Brownian martingales ===
If a [[polynomial]] {{math|''p''(''x'', ''t'')}} satisfies the [[partial differential equation]]
<math display="block">\left( \frac{\partial}{\partial t} + \frac{1}{2} \frac{\partial^2}{\partial x^2} \right) p(x,t) = 0 </math>
then the stochastic process
<math display="block"> M_t = p ( W_t, t )</math>
is a [[martingale (probability theory)|martingale]].

'''Example:''' <math> W_t^2 - t </math> is a martingale, which shows that the [[quadratic variation]] of ''W'' on {{closed-closed|0, ''t''}} is equal to {{mvar|t}}. It follows that the expected [[first exit time|time of first exit]] of ''W'' from (−''c'', ''c'') is equal to {{math|''c''<sup>2</sup>}}.

More generally, for every polynomial {{math|''p''(''x'', ''t'')}} the following stochastic process is a martingale:
<math display="block"> M_t = p ( W_t, t ) - \int_0^t a(W_s,s) \, \mathrm{d}s, </math>
where ''a'' is the polynomial
<math display="block"> a(x,t) = \left( \frac{\partial}{\partial t} + \frac 1 2 \frac{\partial^2}{\partial x^2} \right) p(x,t). </math>

'''Example:''' <math> p(x,t) = \left(x^2 - t\right)^2, </math> <math> a(x,t) = 4x^2; </math> the process
<math display="block"> \left(W_t^2 - t\right)^2 - 4 \int_0^t W_s^2 \, \mathrm{d}s </math>
is a martingale, which shows that the quadratic variation of the martingale <math> W_t^2 - t </math> on [0, ''t''] is equal to
<math display="block"> 4 \int_0^t W_s^2 \, \mathrm{d}s.</math>

About functions {{math|''p''(''xa'', ''t'')}} more general than polynomials, see [[Local martingale#Martingales via local martingales|local martingales]].