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
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
where p(x, t) is the transition density function of {z(t)}. See Williams (1992)
for details.
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Stochastic Processes With Applications To Finance
ISBN: 9781439884829
2nd Edition
Authors: Masaaki Kijima
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