Continuous time one-dimensional asset-pricing models with analytic price–dividend functions

A continuous time one-dimensional asset-pricing model can be described by a second-order linear ordinary differential equation which represents equilibrium or a no-arbitrage condition within the economy. If the stochastic discount factor and dividend process are analytic, then the resulting differential equation has analytic coefficients. Under these circumstances, the one-dimensional Cauchy–Kovalevsky Theorem can be used to prove that the solution to such an asset-pricing model is analytic. Also, this theorem allows for the development of a recursive rule, which speeds up the computation of an approximate solution. In addition, this theorem yields a uniform bound on the error in the numerical solution. Thus, the Cauchy–Kovalevsky Theorem yields a quick and accurate solution of many known asset-pricing models.