Streaming graphs are ubiquitous in daily life, such as evolving social networks and dynamic communication systems. Due to the sensitive information contained in the graph, directly sharing the streaming graphs poses significant privacy risks. Differential privacy, offering strict theoretical guarantees, has emerged as a standard approach for private graph data synthesis. However, existing methods predominantly focus on static graph publishing, neglecting the intrinsic relationship between adjacent graphs, thereby resulting in limited performance in streaming data publishing scenarios. To address this gap, we propose PSGraph, the first differentially private streaming graph synthesis framework that integrates temporal dynamics. PSGraph adaptively adjusts the privacy budget allocation mechanism by analyzing the variations in the current graph compared to the previous one for conserving the privacy budget. Moreover, PSGraph aggregates information across various timestamps and adopts crucial post-processing techniques to enhance the synthetic streaming graphs. We conduct extensive experiments on four real-world datasets under five commonly used metrics. The experimental results demonstrate the superiority of our proposed PSGraph.