As generative AI systems, including large language models (LLMs) and diffusion models, advance rapidly, their growing adoption has led to new and complex security risks often overlooked in traditional AI risk assessment frameworks. This paper introduces a novel formal framework for categorizing and mitigating these emergent security risks by integrating adaptive, real-time monitoring, and dynamic risk mitigation strategies tailored to generative models' unique vulnerabilities. We identify previously under-explored risks, including latent space exploitation, multi-modal cross-attack vectors, and feedback-loop-induced model degradation. Our framework employs a layered approach, incorporating anomaly detection, continuous red-teaming, and real-time adversarial simulation to mitigate these risks. We focus on formal verification methods to ensure model robustness and scalability in the face of evolving threats. Though theoretical, this work sets the stage for future empirical validation by establishing a detailed methodology and metrics for evaluating the performance of risk mitigation strategies in generative AI systems. This framework addresses existing gaps in AI safety, offering a comprehensive road map for future research and implementation.