Drift reconstruction from first passage time data using the Levenberg-Marquardt method

In this paper, we consider the problem of recovering the drift function of a Brownian motion from its distribution of first passage times, given a fixed starting position. Our approach uses the backward Kolmogorov equation for the probability density function (pdf) of first passage times. By taking Laplace transforms, we reduce the problem to calculating the coefficient function in a second-order ordinary differential equation (ODE). The inverse problem effectively amounts to finding the convection coefficient of the ODE, given the transformed pdf for positive values of the Laplace variable. Our first contribution is to find series solutions to the forward problem and show that the associated operator for the linearized inverse problem is compact. Our second contribution is numerical: for low noise levels, we reconstruct simple drift functions by applying Tikhonov regularization and performing a Newton iteration (Levenberg-Marquardt method). For larger noise, our solution displays large oscillations about the true drift.