Fast solutions for the first-passage distribution of diffusion models with space-time-dependent drift functions and time-dependent boundaries

Diffusion models with constant boundaries and constant drift function have been successfully applied to model phenomena in a wide range of areas in psychology. In recent years, more complex models with time-dependent boundaries and space-time-dependent drift functions have gained popularity. One obstacle to the empirical and theoretical evaluation of these models is the lack of simple and efficient numerical algorithms for computing their first-passage time distributions. In the present work we use a known series expansion for the first-passage time distribution for models with constant drift function and constant boundaries to simplify the Kolmogorov backward equation for models with time-dependent boundaries and space-time-dependent drift functions. We show how a simple Crank–Nicolson scheme can be used to efficiently solve the simplified equation.