Two-dimensional diffusion processes as models in lifetime studies

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Let X(t) denote the remaining useful lifetime of a machine, and Y(t) be a standard Brownian motion. Assume that the derivative ρ[X(t), Y(t)] of X(t) is a deterministic function of (at least) Y(t). We consider the two-dimensional degenerate diffusion process (X(t), Y(t)). We obtain explicit expressions for the expected value of the random variable T(x, y) denoting the first time the machine must be replaced, or repaired, for various functions ρ[X(t), Y(t)].