Traditional loss functions, such as mean square error and cross-entropy, treat prediction errors as fixed values, ignoring the stochastic nature of real-world data. To address this limitation, this paper presents three innovative loss function reconstruction methods grounded in stochastic process theory. The first method models prediction errors as stochastic processes by incorporating Gaussian noise and dynamically adjusting the \(\lambda \) parameter, enhancing robustness in noisy environments. The second method introduces a stochastic integral-based loss function for time-series data, representing cumulative error as a time integral of a stochastic process with learnable local volatility parameters to capture temporal dynamics. The third method, randomized cross-entropy (RCE), redefines distribution discrepancy via joint stochastic perturbations to both labels ( \(\eta \sim {\mathcal {N}}(0, \sigma _\eta ^2)\) ) and logits ( \(\xi \sim {\mathcal {N}}(0, \sigma _\xi ^2)\) ), explicitly capturing dual uncertainties from noisy annotations and prediction ambiguity. Experimental results on benchmark datasets (e.g., CIFAR-10, NOAA temperature data) validate that these methods outperform traditional and state-of-the-art baselines (e.g., Label Smoothing, Mixup) in accuracy (up to 2.67% gain) and stability (lower error variance). Despite challenges in computational complexity and parameter tuning, they provide a novel framework for uncertainty-aware machine learning. Source code is available at https://github.com/KaicNK/three_methods.
