The vast quantities of data generated and collected in the Intensive Care Unit (ICU) have given rise to large retrospective datasets that are frequently used for observational studies. The temporal nature and fine granularity of much of the data collected in the ICU enable the pursuit of predictive modeling. In particular, forecasting acute hypotensive episodes (AHE) in intensive care patients has been of interest to researchers in critical care medicine. Given an advance warning of an AHE, care providers may be prompted to search for evolving disease processes and help mitigate negative clinical outcomes. However, the conventionally adopted definition of an AHE does not account for inter-patient variability and is restrictive. To reflect the wider trend of global clinical and research efforts in precision medicine, we introduce a patient-specific definition of AHE in this study and propose deep learning based models to predict this novel definition of AHE in data from multiple independent institutions. We provide extensive evaluation of the models by studying their accuracies in detecting patient-specific AHEs with lead-times ranging from 10 min to 1 hour before the onset of the event. The resulting models achieve AUROC values ranging from 0.57-0.87 depending on the lead time of the prediction. We demonstrate the generalizability and robustness of our approach through the use of independent multi-institutional data.