structural:summarization, quality:correction, information:classification, information:extraction
Time change

If f {\displaystyle f} is an entire function then the process f (Z t) − f (0) {\displaystyle f(Z_{t})-f(0)} is a time-changed complex-valued Wiener process.

Example: Z t 2 = (X t 2 − Y t 2) + 2 X t Y t i = U A (t) {\displaystyle Z_{t}^{2}=\left(X_{t}^{2}-Y_{t}^{2}\right)+2X_{t}Y_{t}i=U_{A(t)}} where A (t) = 4 ∫ 0 t | Z s | 2 d s {\displaystyle A(t)=4\int _{0}^{t}|Z_{s}|^{2}\,\mathrm {d} s} and U {\displaystyle U} is another complex-valued Wiener process.

In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale 2 X t + i Y t {\displaystyle 2X_{t}+iY_{t}} is not (here X t {\displaystyle X_{t}} and Y t {\displaystyle Y_{t}} are independent Wiener processes, as before).