Exercise 12.9 First, confirm that the reflection principle (Proposition 7.2)

holds for any standard Brownian motion. Next, for a standard Brownian motion

{z(t)} with z(0) = x > 0, let M(t) = min0≤u≤t z(u). Using the reflection principle, prove that

Finally, use this equation to obtain (12.35) and (12.36).

Transcribed Image Text:

P{z(t)>y, M(t) 0|2(0) = x} = P{2(t) < y|z(0) = x}, y >0.