Finance, Mathematics, Brownian Motion, Financial Engineering, Mathematical Finance.
condition elaborated. Also, from condition 3 and 4 we know that ∆W(t)=W(t+∆t) - W(t) ~ N(0, ∆t) and each ∆W(t) is iid ~ N(0, ∆t). So, we can easily derive the following formula: Let t = k * ∆t, then W(k) ~ N(0, k) = N(0, k*∆t) . We can also make k approaches infinity to make ∆t approaches 0. Thus,