Network is a simple but powerful representation of real-world complex systems. Network community analysis has become an invaluable tool to explore and reveal the internal organization of nodes. However, only a few methods were directly designed for community-detection in directed networks. In this article, we introduce the concept of local community structure in directed networks and provide a generic criterion to describe a local community with two properties. We further propose a stochastic optimization algorithm to rapidly detect a local community, which allows for uncovering the directional modular characteristics in directed networks. Numerical results show that the proposed method can resolve detailed local communities with directional information and provide more structural characteristics of directed networks than previous methods.