Financial interpretation of Feller’s factorization

The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.