Introductory texts on Wiener Process often introduce the topic by dividing the time into small time steps [tex]h=\Delta t[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Then the value of the process [tex]W_j[/tex] at time j, [tex]t_j[/tex] is given by adding up many independent and normally distributed increments:

[tex]

W_{j+1}=W_j+\sqrt{h}Z_j

[/tex]

The Wiener process [tex]W(t)[/tex] is generated in the limit as the step size [tex]h \rightarrow 0[/tex].

My question is why is the step size [tex]\sqrt{h}Z_j[/tex] and not some other scale of h, such as [tex]h\cdot Z_j[/tex] or [tex]h^2 \cdot Z_j[/tex], say?

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# Brownian Motion

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