Let W be a Brownian motion, is the local time at x, show that the distribution of is an exponential random variable. Determine the parameter of this exponential random variable. Let W be a...

Let W be a Brownian motion,

is the local time at x, show that the distribution of

is an exponential random variable. Determine the parameter of this exponential random variable.

Let W be a one-dimensional Brownian motion. This exercise asks you to prove that the normalized number of down crossings by time t converges to local time at 0. If

Then Dt(a), the number of down crossings up to time t, is defined to be

Prove that there exists a constant c such that

Where

is local time at 0 of W. Determine c.