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.
Highlights
A goal of this work, complementary to our recent paper [1], is to elaborate on the Cauchy initial value problem for a class of nonautonomous and inhomogeneous diffusion-type equations on R
Ermakov)-type systems, which seem to be missing in the available literature
A group theoretical approach to a similar class of partial differential equations is discussed in Refs. [14,15,16]
Summary
A goal of this work, complementary to our recent paper [1], is to elaborate on the Cauchy initial value problem for a class of nonautonomous and inhomogeneous diffusion-type equations on R. A group theoretical approach to a similar class of partial differential equations is discussed in Refs. If p (x, y, t) is the appropriate fundamental solution of Equation (3), one can compute the given expectations according to In this context, the fundamental solution is known as the probability transition density for the process and p (x, y, t) dy = 1.
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