I got some problems with this homework which I have totally no idea, never got into this field before and I really need some help.

First, we have a wiener process like

Which means the probability of the process drops beneath -3 within the time interval [0,1].

Now the thing is we have to simulate the process by discretize it.

1.Suppose we first discretize the process by 100 points and simulate 10,000 process in this way.

i.e., W(0.01), W(0.02), …., W(1.00).

Note that W(t) – W(t-0.01) ~ N(0,0.01) independently.

2.Using the data obtained at 1., we approximate

by

what is the relationship between this value and the real

(larger, equal to or smaller)?

3.Repeat 1. and 2. by cutting [0,1] into 10,000 points instead. Will the resulting probability increases or decreases?