The Path-Dependent Neural Jump Ordinary Differential Equation (PD-NJ-ODE) is a model for predicting continuous-time stochastic processes with irregular and incomplete observations. In particular, the method learns optimal forecasts given irregularly sampled time series of incomplete past observations. So far the process itself and the coordinate-wise observation times were assumed to be independent and observations were assumed to be noiseless. In this work we discuss two extensions to lift these restrictions and provide theoretical guarantees as well as empirical examples for them. In particular, we can lift the assumption of independence by extending the theory to much more realistic settings of conditional independence without any need to change the algorithm. Moreover, we introduce a new loss function, which allows us to deal with noisy observations and explain why the previously used loss function did not lead to a consistent estimator.

翻译：路径依赖神经跳跃常微分方程（PD-NJ-ODE）是一种用于预测具有不规则和不完全观测的连续时间随机过程的模型。具体而言，该方法能在给定非规则采样的不完全历史观测时间序列时，学习最优预测。目前，该模型假设过程本身与逐坐标观测时间相互独立，且观测数据无噪声。本文提出两项扩展以解除这些限制，并为这些扩展提供理论保证及实证案例。具体而言，我们通过将理论扩展至更具现实意义的条件独立场景，无需修改算法即可解除独立性假设。此外，我们引入一种新的损失函数以处理噪声观测，并解释为何先前采用的损失函数无法生成一致估计量。