As more and more AI agents are used in practice, it is time to think about how to make these agents fully autonomous so that they can learn by themselves in a self-motivated and self-supervised manner rather than being retrained periodically on the initiation of human engineers using expanded training data. As the real-world is an open environment with unknowns or novelties, detecting novelties or unknowns, characterizing them, accommodating or adapting to them, gathering ground-truth training data, and incrementally learning the unknowns/novelties are critical to making the agent more and more knowledgeable and powerful over time. The key challenge is how to automate the process so that it is carried out on the agent's own initiative and through its own interactions with humans and the environment. Since an AI agent usually has a performance task, characterizing each novelty becomes critical and necessary so that the agent can formulate an appropriate response to adapt its behavior to accommodate the novelty and to learn from it to improve the agent's adaptation capability and task performance. The process goes continually without termination. This paper proposes a theoretic framework for this learning paradigm to promote the research of building Self-initiated Open world Learning (SOL) agents. An example SOL agent is also described.