Wiener Process

[Jcrew]

So a Wiener process can be modeled by the following difference equation:

W(t) = W(t-1) + N(0,1)

where N(0,1) = normal distribution with mean 0 and unit variance 1.

Somewhere I saw someone use the following closed form for it:

W(t)= sqrt(t) * N(0,1)

Is the closed form an approximation for the difference equation? If not can someone show the steps to reach to the close form from the difference equation?

[ncray]

If W(0) = 0, then W(1) = X_1, where X_1 ~ N(0,1), and W(t) = X_1 + ... + X_t where X_i ~ N(0,1). The sum of t iid normal RVs is normal with mean 0 and variance t, so W(t) ~ N(0,t) which is the same as sqrt(t) * N(0,1)

[Jcrew]

Thanks. Can you explain how you get from N(0,t) ~ sqrt(t) * N(0,1)