Schedule

From Monday, 11 September 2023 till Friday, 15 September 2023: – Morning: 9.30-11 and 11.30-13.00; – Afternoon: 14.30-16.00 and 16.30-18

Morning Lectures will be mainly devoted to theory and examples. Afternoon Lectures will be mainly devoted to questions and exercises in an interactive way: for example the teacher will suggest some exercises to do leaving some time to do them and then discuss them at the blackboard.

Outline of the course Day 1: Introduction to the Dynamic Optimization in discrete and continuous time; Dynamic Programming in discrete time. - Examples of Dynamic Optimization problems in Economics: - Basic ones: utility maximization, optimal investment, optimal portfolio; - Recent ones: climate change and economics, control of infectious diseases, pollution. - Dynamic optimization problems as Optimal Control Problems: mathematical setting in discrete and continuous time: state equation, pointwise constraints, set of admissible control strategies, objective functional, optimal strategies, optimal state paths, value function. - Discounted autonomous infinite horizon problems. - Feedback control strategies: admissible and optimal feedback maps. - Dynamic Programming (DP) Principle and Bellman equation in discrete time, finite and infinite horizon. - Optimality conditions via DP: verification theorem. - Guess-and-verify method and examples. - Solution of Bellman equation through fixed point theorems.

Day 2: Dynamic Programming in continuous time. HJB equations and viscosity solutions. - Dynamic Programming in continuous time: value function, dynamic programming principle, HJB equation: the finite horizon case and the infinite horizon autonomous case with discount. Examples. - Optimality conditions via DP: verification theorem. - Guess-and-verify method and examples. - Some basics on Viscosity Solutions Theory for HJB equations and Examples.

Day 3: Maximum Principle in discrete and continuous time. - Pontryagin Maximum Principle (PMP) in discrete time: the finite horizon case. - PMP in the infinite horizon case: transversality conditions. Examples. - PMP in continuous time, finite horizon. - PMP in continuous time, infinite horizon. Transversality conditions. Saddle path stability. - Examples of applications.

Day 4: Existence and uniqueness of optimal strategies Some basic tools of functional analysis: Banach and Hilbert spaces, weak topologies and convexity. Existence theorems trough compactness. Uniqueness theorems through strict convexityconcavity or through verification theorems

Day 5: Introduction to stochastic control in discrete and continuous time Motivating examples. Some ideas on controlled Stochastic DifferenceDifferential equations The discrete time case: Bellman equations and verification theorem The continuous time case: Ito formula, HJB equations and verification theorem. Application to optimal portfolio problems

Main exercises of the course - Check if a given control strategy is admissible - Check if a given feedback map is admissible - Given a Dynamic Optimization problem translate it as an Optimal Control Problem in standard form. - Prove that the value function is finite. - Given a Dynamic Optimization problem in discretecontinuous time write the necessary conditions of the maximum principle and, in simple cases, try to solve them. - Solve simple Dynamic Optimization problems in discretecontinuous time and in finiteinfinite horizon with the Dynamic Programming method, writing the BellmanHJB equation and solving it.