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Abstract
To model the remaining lifetime X(t) of machines, two-dimensional degenerate diffusion processes (X(t), Y(t)) are considered. The process Y(t) is assumed to be a geometric Brownian motion, while the derivative of X(t) is a negative deterministic function of X(t) and Y(t). Hence, the process X(t) is decreasing. We calculate explicitly for a class of processes the expected value of the random variable that denotes the time it takes X(t) to reach c, given that X(0) = x > c and Y(0) = y > 0. The method of similarity solutions is used to solve the appropriate Kolmogorov backward equation.
Published Version
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