Question: If the Brownian motion is stopped at the moment it reaches a for the first time, the process y(t) = w(t) for t ≤ τa and y(t) = a for t ≥ τa is called the absorbed Brownian motion.
prove that the pdf of y(t), denoted fy(·)(x, t), satisfies for x < a="" the="" diffusion="" equation="" and="" the="" initial="" condition="" (2.26)="" and="" the="" boundary="" condition="">
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Assume that the trajectories of the MBM begin at time t0 at a point x0 < a.="" find="" the="" pdf="" of="" the="" absorbed="" mbm="" for="" this="" case.="">
(iii) Prove that the pdf of the absorbed MBM in (ii), denoted , is the solution of the initial and boundary value problem for the diffusion equation
