July 28th, 2009 Cho, Maxx. & Glasgow, Scott. "Stochastic Interest Rate Distributions via Green's Functions".
The Term Structure Interest Rate Model of Cox, Ingersoll, and Ross (CIR) is a commonly used model when considering interest rates as a random variable. However, the CIR stochastic differential equation has no known closed-form solutions. In this paper, a Green’s function approach to solving the differential equation not commonly employed outside of mathematical physics is used to derive a closed-form formula for the density function of the interest rate process that agrees with the distribution Cox, Ingersoll, and Ross originally discovered. Our formulation of this distribution is entirely rigorous and logically sound, providing an ansatz-free derivation.