Argument Component Boundary Detection (ACBD) is an important sub-task in argumentation mining; it aims at identifying the word sequences that constitute argument components, and is usually considered as the first sub-task in the argumentation mining pipeline. Existing ACBD methods heavily depend on task-specific knowledge, and require considerable human efforts on feature-engineering. To tackle these problems, in this work, we formulate ACBD as a sequence labeling problem and propose a variety of Recurrent Neural Network (RNN) based methods, which do not use domain specific or handcrafted features beyond the relative position of the sentence in the document. In particular, we propose a novel joint RNN model that can predict whether sentences are argumentative or not, and use the predicted results to more precisely detect the argument component boundaries. We evaluate our techniques on two corpora from two different genres; results suggest that our joint RNN model obtain the state-of-the-art performance on both datasets.