Simulation of the Wiener process

Question

While simulating the standart Wiener process one should use the expression $$W(t+\Delta t)-W(t)=\sqrt{\Delta t}\, \xi_t, $$ where $\xi_t \sim N(0,1)$. Explain me, please, how has the multiplier $\sqrt{\Delta t}$ come up?

Answer 1

Recall that $W(t + \Delta t) - W(t) \sim N(0, \Delta t)$. If $\xi_t \sim N(0,1)$ then $\sqrt{\Delta t} \, \xi_t \sim N(0,\Delta t)$