Prove that the hitting time distribution of a Wiener process (starting in c > 0, with µ = 0 and σ = 1) is the inverse Gaussian distribution (10.2), using the following arguments. Recall that W(t) has...

Prove that the hitting time distribution of a Wiener process (starting in c > 0, with µ = 0 and σ = 1) is the inverse Gaussian distribution (10.2), using the following arguments. Recall that W(t) has a Gaussian distribution with mean c and variance t. Let T be the first time W(t) hits zero, but assume that W is not absorbed there. Note first that
 Argue informally that

  Use this to prove that
 and derive the density of T. More details can be found in Karatzas and Shreve (1991).