Continuous-time perpetuities and time reversal of diffusions
The authors discuss the estimation of the joint distribution of a continuous-time perpetuity and some underlying factors governing the cash flow rate under an ergodic Markovian model. They consider two approaches to the problem, namely the partial differential equation approach and the time reversal approach. In the partial differential equation approach, the authors derive a partial differential equation for the conditional cumulative distribution of the perpertuity given the initial values of the factors. In the time reversal approach, they obtain the joint probability law as the stationary distribution of an ergodic multi-dimensional diffusion. Section 2 describes the problem. The main results including the partial differential equation for the cumulative distribution function and the joint probability law based on the time reversal diffusion are presented in Section 3. In Section 4, the authors present a numerical example for the time reversal approach based on Monte Carlo simulation.
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