Data-driven uncertainty quantification for constrained stochastic differential equations and application to solar photovoltaic power forecast data

In this work, we extend the data-driven Itô stochastic differential equation (SDE) framework for the pathwise assessment of short-term forecast errors to account for the time-dependent upper bound that naturally constrains the observable historical data and forecast. We propose a new nonlinear and time-inhomogeneous SDE model with a Jacobi-type diffusion term for the observable phenomenon of interest, simultaneously driven by the forecast and the constraining upper bound. We rigorously demonstrate the existence and uniqueness of a strong solution to the SDE model by imposing a condition for the time-varying mean-reversion parameter appearing in the drift term. After normalization, the original forecast function is thresholded to keep such time-varying mean-reversion parameters bounded. Thus, for any finite time interval, the paths of the forecast error process almost surely do not reach the time-dependent boundaries. The SDE model parameter calibration procedure is applied to user-selected approximations of the likelihood function. Another novel contribution is estimating the unknown transition density of the forecast error process with a tailored kernel smoothing technique without and with a control variate method, coupling an adequate SDE to the original one. We provide the theoretical study about how to choose the optimal bandwidth. As a case study, we fit the model to the 2019 photovoltaic (PV) solar power daily production and forecast data in Uruguay, computing the daily maximum solar PV production estimation. Two statistical versions of the constrained SDE model are fit, with the beta and truncated normal distributions as surrogates for the transition density function of the forecast error process. Empirical results include simulations of the normalized solar PV power production and pathwise confidence bands generated with the desired coverage probability through an indirect inference method. An objective comparison of optimal parametric points associated with the two selected statistical approximations is provided by applying our innovative kernel smoothing estimation technique of the transition function of the forecast error process. As a byproduct, we created a procedure providing a reliable criterion for choosing an adequate density proxy candidate that better fits the actual data at a low time cost. The methodology employs a thorough pathwise assessment of the forecast error uncertainty in situations where time-dependent boundaries and available forecasts drive the model specifications.