Foundations of Stochastic Processes: Probability and Modeling

Understanding randomness is crucial for modern quantitative roles in finance, risk management, and systems engineering. This course demystifies the complex mathematics behind random systems, turning abstract probability theory into practical analytical skills. You will transition from basic probability concepts to understanding advanced stochastic models, including martingales, Brownian motion, and stochastic integration. By reading our structured explanations and working through written exercises, you will gain the confidence to apply these mathematical models to real-world scenarios. What you'll learn: - Understand foundational measure-theoretic probability and key terminology of random variables. - Analyze martingales, filtrations, and stopping times to model sequential decision-making. - Explore Brownian motion and its mathematical properties for continuous-time modeling. - Apply Ito calculus and stochastic integration to financial mathematics and derivative pricing. - Study functional limit theorems and large deviations to understand long-term system behaviors. - Evaluate modern applications in queueing theory, inventory models, and algorithmic risk analysis. The course begins with foundational definitions of probability spaces and random variables, gradually progressing through martingales and continuous-time processes, and concluding with practical applications in finance and operations. Designed for beginners in quantitative analysis, finance, or data science, this course requires only a basic background in calculus and algebra, with no prior advanced probability knowledge needed. Begin your journey into stochastic modeling today.