The Riccati System and a Diffusion-Type Equation

Abstract

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.

Keywords

• diffusion-type equations
• Green’s function
• fundamental solution
• autonomous and nonautonomous Burgers equations
• Fokker-Planck equation
• Black-Scholes equation
• the Hull-White model
• Riccati equation and Riccati-type system
• Ermakov equation and Ermakov-type system