Department of Mathematics and Industrial Engineering, Polytechnique Montreal, Montreal, Canada
Abstract
Let X(t) denote the remaining lifespan of a device. Two-dimensional degenerate diffusion processes (X(t),Y(t)), where Y(t) is a variable that influences the remaining lifespan, are proposed to model the evolution of X(t) over time. These processes are defined in such a way that X(t) will be strictly decreasing as time increases: dX(t) = rho[X(t),Y(t)] dt, where rho is a strictly negative function and {Y(t), t > 0} is a diffusion process. Next, optimal control problems for such diffusion processes, in which the final time is the random time when the device is considered to be worn out, are considered. This type of problems is known as homing problems. The dynamic programming equation satisfied by the value function is derived and particular problems are solved explicitly for diffusion processes {Y(t), t > 0} that are important for applications. To do so, we must solve non-linear partial differential equations, subject to the appropriate boundary conditions.
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