<p>Live streaming has become an increasingly critical sales channel in e-commerce, although the conversion rate from viewership to actual purchases remains relatively low. To better understand and model consumer decision-making processes, this paper presents a systematic framework for time-to-purchase prediction by formalizing fine-grained user viewing behaviors and incorporating an enhanced Hawkes process into an encoder-decoder architecture. The core innovation of this work lies in the proposed ODE-LSTM Hawkes model, which leverages neural ordinary differential equations to capture continuous-time behavioral dynamics, thereby alleviating the limitations of traditional static intensity functions. By adapting methodologies from financial event analysis to the live streaming scenario, the framework unifies multi-type user interactions including viewing, bookmarking, and purchasing within a multivariate Hawkes process. In addition, we introduce a new engagement efficiency metric derived from watch-to-order time and enable the joint prediction of action types and their corresponding timestamps. Extensive experimental results on real-world e-commerce datasets verify that the proposed model achieves superior performance over existing state-of-the-art methods in forecasting user viewing and purchasing behaviors.</p>