Simulation of Gaming Strategies Using Martingales and Stopping Times in Stochastic Processes

This major assignment requires applying knowledge of stochastic processes, martingales, and stopping times to simulate gaming strategies. As part of Tsinghua University's Electronic Engineering curriculum, it demands thorough code documentation and fully debugged implementations. The assignment begins with understanding fundamental concepts of stochastic processes, martingales, and stopping time simulations. We first explore the definition of stochastic processes and their applications in probability theory, examining how they model real-world scenarios through sequential random variables. The implementation typically involves generating random paths using probability distribution functions and time-step simulations. Next, we delve into martingale theory, analyzing its properties (such as the conditional expectation property E[X_{n+1}|X_n] = X_n) and applications in probability modeling. Code implementation often includes creating martingale sequences with proper filtering mechanisms and validation checks to ensure the martingale property holds throughout simulations. Finally, we study stopping time strategies in game simulations, focusing on their practical applications for decision-making points in stochastic processes. The coding aspect involves implementing conditional checks and boundary conditions that determine optimal stopping rules based on predefined criteria. Beyond theoretical understanding, the assignment requires practical programming skills to implement these concepts. This includes using tools like MATLAB for stochastic path simulations, coding martingale sequences with proper error handling, and developing stopping time algorithms with exit conditions. The implementation typically involves: - Random number generation with seed control for reproducible results - Matrix operations for efficient path calculations - Conditional loop structures for stopping time evaluations - Statistical analysis functions for result validation Substantial time must be dedicated to hands-on practice, including code development, debugging sessions, and performance optimization. The debugging phase particularly focuses on verifying probability conservation in martingales and ensuring stopping conditions trigger correctly. In summary, this assignment demands mastery of stochastic process theory, martingale properties, and stopping time strategies, combined with strong programming capabilities. Through completion, students gain deep understanding of these concepts and learn to apply them to practical problem-solving scenarios with robust computational implementations.