Advanced topics in the theory and application of probability and statistics. Recent topics include nonparametric statistics and advanced probability. Study Ito-diffusion processes, stochastic calculus, and stochastic integration. Explore stochastic differential equations and their connection to classical analysis. Discuss an introduction to optimal stochastic control of diffusion processes, the Hamilton-Jacobi-Bellman equation (classical and viscosity solutions), singular stochastic control, and linear filtering. Present applications, mainly from mathematical finance, inventory theory, decision analysis, and insurance. Examine brief overview of multi-scale problems in stochastic analysis.