We develop methods for the approximation of solutions to the Chapman-Kolmogorov backward and Feynman-Kac partial di.erential equations, where the method of approximation is accurate for very long time horizons. When an underlying economy is modeled by a di.usion process, asset prices and conditional expectations of the state variables can be found as solutions to these partial di.erential equations. However, for all but a few simple cases, solutions cannot be found explicitly in closed form. The form of these equations suggests constructing a power series in the time variable as a method of solution. However, the convergence properties of such power series solutions are often quite poor. We examine the problem of determining the convergence properties of power series solutions, and introduce a parameterized family of non-a.ne transformations of the time variable that can substantially improve the rate of convergence for long time horizons. In some cases, the approximations converge uniformly (in time) to the true (but unknown) solutions for arbitrarily large time horizons. The ability to approximate solutions accurately and in closed form simplifies the estimation of non-a.ne continuous-time term structure models, since the bond pricing problem must be solved for many di.erent parameter vectors during a typical estimation procedure.
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