I. MATHEMATICAL PRELIMINARIES
Chapter 1: Stochastic processes, Brownian motion, diffusions (08-04-04)
1. Random variables, stochastic processes
2. Independence
3. Wiener processes, Brownian motion
4. Random walk approximation of Brownian motion
5. Stopping times
6. Strong Markov property
7. Diffusions
8. Discrete approximation of Ornstein-Uhlenbeck processes
Chapter 2: Stochastic integrals, Ito’s lemma (08-04-04)
1. The Hamilton-Jacobi-Bellman equation
2. Stochastic integrals
3. Ito’s lemma
4. Geometric Brownian motion
5. Occupancy measure, local time
6. Tanaka’s formula
Chapter 3: Martingales (08-06-04)
1. Definition, examples
2. Martingales based on eigenvalues
3. The Wald martingale
4. Sub- and supermartingales
5. Optional stopping theorem
6. Optional stopping theorem, extended
Chapter 4: Useful formulas for Brownian motions (02-04-05/)
1. Stopping times defined by thresholds
2. Expected values for Wald martingales
3. Discounted values for hitting times
4. ODEs for Brownian motions
5. Solutions for r = 0
6. Discounted values: solutions for r > 0
7. Expected returns over an infinite horizon
8. ODES for diffusions
9. Solutions for r = 0
10. Solutions for r > 0
11. Discounted local time
II. IMPULSE CONTROL MODELS
Chapter 5: Exercising an option (09-20-04)
1. The determistic problem
2. The stochastic problem: a direct approach
3. Using the Hamilton-Jacobi-Bellman equation
4. An example
Chapter 6: Menu cost models (12-07-04)
1. The model
2. Reformulating the problem
3. A direct approach
4. Using the Hamilton-Jacobi-Bellman equation
5. Calculating the solution
Chapter 7: Aggregate models with price adjustment (02-21-05)
1. The economic environment
2. An economy with monetary neutrality
3. An economy with a Phillips curve
4. Optimizing behavior and the Phillips curve
5. Money, prices and output
6. Motivating the pricing rule
7. Appendix
Chapter 8: Inventory models (03-11-05)
1. The basic model
2. Is control exercised?
3. Characterizing the optimal policy
4. Obtaining the solution from the HJB equation
5. The stationary distribution
6. The cash management problem
7. An inventory problem with b = 0
8. Strictly convex adjustment costs
Chapter 9: Durable goods (04-14-05)
1. No transaction costs
2. The model with transaction costs
3. Reformulating the problem
4. A special case: ² = 1 and fixed a
5. Using the HJB equation
6. Adding portfolio choice
7. Exercises
III. INSTANTANEOUS CONTROL MODELS
Chapter 10: Regulated Brownian motion (09-10-03)
1. The one-sided regulator
2. The two-sided regulator
3. Discounted values
4. The stationary distribution
5. An inventory example
Chapter 11: Investment (04-07-05)
1. A unified model of investment
2. Abel and Eberly’s “exact solution”
3. Irreversible investment: one shock
4. Two shocks: (Bertola and Caballero)
Appendix A:
1. Modes of convergence
2. Continuous stochastic process
3. Wiener measure
4. Total and quadratic variation
Appendix B: Stochastic integration
1. Integrals
2. Preliminaries
3. Stochastic integrals of simple processes
4. Approximating sequences
5. Stochastic integrals
6. Example: the integral of a Wiener process
7. Ito processes, Ito’s lemma
Appendix C: Martingales
1. Properties of martingales
2. Optional stopping theorem
3. Crossing results
4. Martingale convergence theorem | 