Editing Wiener process (section)

=== Covariance and correlation ===
The [[covariance function|covariance]] and [[correlation function|correlation]] (where <math>s \leq t</math>):
<math display="block">\begin{align}
\operatorname{cov}(W_s, W_t) &= s, \\
\operatorname{corr}(W_s,W_t) &= \frac{\operatorname{cov}(W_s,W_t)}{\sigma_{W_s} \sigma_{W_t}} = \frac{s}{\sqrt{st}} = \sqrt{\frac{s}{t}}.
\end{align}</math>

These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that <math>t_1\leq t_2</math>.
<math display="block">\operatorname{cov}(W_{t_1}, W_{t_2}) = \operatorname{E}\left[(W_{t_1}-\operatorname{E}[W_{t_1}]) \cdot (W_{t_2}-\operatorname{E}[W_{t_2}])\right] = \operatorname{E}\left[W_{t_1} \cdot W_{t_2} \right].</math>

Substituting
<math display="block"> W_{t_2} = ( W_{t_2} - W_{t_1} ) + W_{t_1} </math>
we arrive at:
<math display="block">\begin{align}
\operatorname{E}[W_{t_1} \cdot W_{t_2}] & = \operatorname{E}\left[W_{t_1} \cdot ((W_{t_2} - W_{t_1})+ W_{t_1}) \right] \\
& = \operatorname{E}\left[W_{t_1} \cdot (W_{t_2} - W_{t_1} )\right] + \operatorname{E}\left[ W_{t_1}^2 \right].
\end{align}</math>

Since <math> W_{t_1}=W_{t_1} - W_{t_0} </math> and <math> W_{t_2} - W_{t_1} </math> are independent,
<math display="block"> \operatorname{E}\left [W_{t_1} \cdot (W_{t_2} - W_{t_1} ) \right ] = \operatorname{E}[W_{t_1}] \cdot \operatorname{E}[W_{t_2} - W_{t_1}] = 0.</math>

Thus
<math display="block">\operatorname{cov}(W_{t_1}, W_{t_2}) = \operatorname{E} \left [W_{t_1}^2 \right ] = t_1.</math>

A corollary useful for simulation is that we can write, for {{math|''t''<sub>1</sub> < ''t''<sub>2</sub>}}:
<math display="block">W_{t_2} = W_{t_1}+\sqrt{t_2-t_1}\cdot Z</math>
where {{mvar|Z}} is an independent standard normal variable.