In mathematics, continuous martingales and Brownian motion are important concepts that have numerous applications in finance, physics, and engineering. A continuous martingale is a stochastic process that models the evolution of a system that adjusts to new information over time. Brownian motion, on the other hand, describes the random movement of a particle in a fluid or gas. In this article, we will explore these concepts in detail and provide examples of their real-world applications. Whether you are a student of mathematics or a professional seeking to deepen your understanding of stochastic processes, this article will provide valuable insights.