A Stochastic Process Perspective on Hybrid Log-Normal and Machine Learning Models for Financial Risk under Left-Censored Data

In the context of financial risk management, predictive modeling under censored data remains a complex challenge. This paper develops and compares two approaches: a traditional log-normal regression model and a hybrid framework combining log-normal regression with an XGBoost-based correction layer. While the parametric component captures the structured relationships between covariates and claim costs, the machine learning layer adjusts for nonlinear residual structure. Building on this, we introduce a stochastic interpretation of the hybrid estimator by modeling prediction errors as a Gaussian process. We derive a formal variance decomposition, separating model-based and correction-layer uncertainty. To quantify this, we implement both simulation-based estimation and diagnostic tools for residual stationarity and ergodicity. Additionally, we propose a Bayesian stochastic extension by placing priors over model parameters and deriving posterior predictive intervals. A novel contribution of this work is the incorporation of residual dynamics via autoregressive stochastic processes, where residuals from the hybrid model are modeled as AR(۱) processes and also as a Diffusion Process. This allows for modeling temporal dependence and improves interpretability of correction structures.

کلیدواژه ها:

log-normal regression ، hybrid model ، Hybrid estimator ، censored ، stochastic process ، Value at Risk