Lecture
This lecture covers the principles of Monte Carlo methods for estimating expectations, variance reduction techniques, and stochastic optimal control. It delves into the theoretical foundations, confidence intervals, mean square error, efficiency of simulation estimators, optimal allocation of computing time, and numerical results. The lecture also explores the Black-Scholes stock model, different simulation schemes, control variates, importance sampling, and the efficiency of importance sampling. Additionally, it discusses the stochastic optimal control problem, the Hamilton-Jacobi-Bellman equation, investment and consumption dynamics, and the optimal allocation of investments and consumption. The lecture concludes with the derivation of the Hamilton-Jacobi-Bellman equation and the verification theorem.