Inverse problems for a class of stochastic differential equations in a generalized fiducial setting.

Abstract: In this talk, we approach the inverse problem of parameter inference for stochastic ordinary differential equations with constant drift. The data for inference comes not as a direct observation of the process but time averages against some test function. We frame the inverse problem in a generalized fiducial framework. However, it is difficult to invert the data-generating equation directly and get a solution. As a work-around, we approximate the data-generating equation by a piece-wise linear function over a partition of the parameter space. We show that as the partition cells shrink, the approximation uniformly approaches the true function almost surely.

For this approximation, we study when solutions to the inversion problem actually exist by examining the distributions of the linearization coëfficients using tools from Malliavin calculus. In the regime k ≤ (p + 1) (more parameters than observations), solutions exist almost surely, and we get their empirical fiducial distribution by sampling from level sets of the approximation at the observed value. In the case k > (p + 1) (fewer parameters than observations), this would not work since almost surely no solution exists to the problem. Following the usual approach in generalized fiducial inference, we condition on a solution existing. Sampling from this conditional distribution is difficult. So, we further approximate this conditional distribution using a Normal in what we term variational fiducial inference. In both regimes, we demonstrate the efficacy of our method through numerical examples.

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