On MAP estimates and source conditions for drift identification in SDEs
Theoretical and numerical analysis of MAP estimation for drift identification in SDEs, an incremental contribution to inverse problems for stochastic processes.
The paper derives a MAP estimate for drift identification in SDEs from discrete observations, proves differentiability and a tangential cone condition, and provides numerical evidence suggesting convergence rates as n→∞.
We consider the inverse problem of identifying the drift in an SDE from $n$ observations of its solution at $M+1$ distinct time points. We derive a corresponding MAP estimate, we prove differentiability properties as well as a so-called tangential cone condition for the forward operator, and we review the existing theory for related problems, which under a slightly stronger tangential cone condition would additionally yield convergence rates for the MAP estimate as $n\to\infty$. Numerical simulations in 1D indicate that such convergence rates indeed hold true.