Part I: Convex extensions of the Markowitz portfolio selection problem

• Basic notions from convex optimization
• Criteria of convexity
• Classes of convex problems: LP, SOCP, SDP
• Duality
• Vector optimization: Pareto optimal solutions, Scalarization
• Introduction to the CVX solver (as extension to Matlab) as special solver for
• convex optimization problems
• Examples from finance and their numerical solution via CVX:
• Extensions of Markowitz portfolio selection problem and their convexification: incomplete information about covariance, probability constraints, robust optimization
• Factor models
• Log-optimal investment strategy
• Sharpe ratio

Part II: Risk measures and decision models

• Basic notions and concepts
• Single-stage decision models
• Risk deviation functionals and their properties
• Standard and less standard risk measures
• More on Conditional value-at-risk
• Formulation of decision optimization models with different risk measures as objectives and their transformation to LP
• Multi-stage decision models
• Multi-stage risk deviation functionals
• Formulation of multi-stage decision models, their transformation to linear programs, large-scale optimization

Part III: Hamilton-Jacobi-Bellman equation for dynamic portfolio optimization problems

• Expected utility maximization problem
• Bellman's optimality principle and its application to the utility maximization problem
• Deriving a Hamilton-Jacobi-Bellman (HJB) nonlinear partial differential equation
• Notes on standard ways of solving HJB equations
• Riccati-type of transformation to HJB equations
• Advantages of the transformed HJB equations
• Worst-case portfolio optimization using techniques from Part I
• Inter-temporal utility optimization in the HJB context