Exercise 12.11 Consider a Brownian motion {X(t)} with drift and diffusion coefficient . Suppose that X(0)

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Exercise 12.11 Consider a Brownian motion {X(t)} with drift μ and diffusion coefficient σ. Suppose that X(0) = 0, and let τ denote the first time that the Brownian motion reaches the state x > 0. Prove that the density function for τ is given by

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Note: For a standard Brownian motion {z(t)}, let τ be the first passage time to the boundary a(t) > 0. If the derivative a′(t) exists and is continuous, then the density function f(t) for τ satisfies the equation

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where p(x, t) is the transition density function of {z(t)}. See Williams (1992)
for details.

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