In this work, we extend the data-driven It\^{o} 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 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. The normalized forecast function is thresholded to keep such mean-reversion parameters bounded. The SDE model parameter calibration 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 with the control variate method, coupling an adequate SDE to the original one. We provide a theoretical study about how to choose the optimal bandwidth. 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 proxies for the transition density. Empirical results include simulations of the normalized solar PV power production and pathwise confidence bands generated 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.

翻译：本研究扩展了数据驱动的伊藤随机微分方程框架，用于短期预测误差的路径评估，以考虑自然约束观测历史数据与预测结果的时变上界。我们针对目标现象提出了一种新型非线性非齐次随机微分方程模型，该模型采用雅可比型扩散项，并同时受预测函数和约束上界驱动。通过为漂移项中时变均值回归参数施加约束条件，我们严格证明了该随机微分方程模型强解的存在唯一性。归一化预测函数经过阈值处理以保持此类均值回归参数的有界性。模型参数校准采用用户选择的似然函数近似方法。另一创新贡献在于：通过将适配的随机微分方程与原方程耦合，采用控制变量法的定制核平滑技术估计预测误差过程未知的转移密度。我们提供了关于最优带宽选择的理论研究。该模型应用于乌拉圭2019年光伏太阳能日发电量及预测数据，计算了日最大光伏发电量估计值。我们拟合了约束随机微分方程模型的两种统计版本，分别以贝塔分布和截断正态分布作为转移密度的代理分布。实证结果包括归一化光伏发电功率的模拟数据，以及通过间接推断方法生成的路径置信带。通过应用我们创新的预测误差过程转移函数核平滑估计技术，对两种选定统计近似方法对应的最优参数点进行了客观比较。