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 also covers the thresholding parameter of the normalized forecast by applying a novel iterative two-stage optimization procedure to user-selected approximations of the likelihood function. Another novel contribution is estimating the transition density of the forecast error process, not known analytically in a closed form, through a tailored kernel smoothing technique with the control variate method. 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 the innovative kernel density estimation technique of the transition function of the forecast error process.

翻译：本文扩展了数据驱动伊藤随机微分方程（SDE）框架，用于路径依赖的短期预测误差评估，以考虑随时间变化的上界约束——该上界自然限制了可观测历史数据与预测值。我们针对感兴趣的现象提出一种新型非线性非时齐SDE模型，该模型包含雅可比型扩散项，并同时受预测值与约束上界驱动。通过施加漂移项中时变均值回归参数的有界条件，我们严格证明了该SDE模型强解的存在唯一性。归一化预测函数经阈值化处理以保持均值回归参数的有界性。SDE模型参数标定过程通过创新性的迭代两阶段优化方法，对用户选定的似然函数近似形式进行优化，从而覆盖归一化预测的阈值参数。另一项创新贡献在于：针对无法解析封闭求解的预测误差过程转移密度，采用带控制变量法的定制核平滑技术进行估计。我们将模型拟合至乌拉圭2019年光伏（PV）日发电量预测数据，计算日最大光伏发电量估计值。分别以贝塔分布和截断正态分布作为转移密度的代理，拟合了两种统计版本的约束SDE模型。实证结果包括通过间接推断方法生成的归一化光伏发电量模拟及其路径依赖置信带。通过应用预测误差过程转移函数的创新性核密度估计技术，提供了与两种选定统计近似对应的最优参数点的客观比较。