Kramers–Moyal expansion for stochastic differential equations with single and multiple delays: Applications to financial physics and neurophysics

We present a generalized Kramers–Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker–Planck equation derived earlier in the literature is a special case of the proposed Kramers–Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed.