ur goal is to compute the probability that the standard Brownian motion
hits a given level
before time reaches
.
We are going to use the (
Reflection
principle
). First, we note that the condition
of the reflection principle is essential. Indeed, suppose we forget about it
and
write
Therefore, in order to use the reflection principle correctly, we split values
into the two
intervals:
We perform the change of variable
in the second
integral.
This result may be understood in line of the proof of the the reflection
principle. For every path that hits the level
and ends below
there is a reflection that hits the level
and ends symmetrically above
.
Hence, one
comes directly from those scenarios when the path goes to
,
and another
comes from the inverted paths.
