Time-dependent solutions for stochastic systems with delays: Perturbation theory and applications to financial physics

First-order approximations of time-dependent solutions are determined for stochastic systems perturbed by time-delayed feedback forces. To this end, the theory of delay Fokker–Planck equations is applied in combination with Bayes' theorem. Applications to a time-delayed Ornstein–Uhlenbeck process and the geometric Brownian walk of financial physics are discussed.