Undervisningsplan

• optimization for any single agents in the market

• definition of equilibrium

• characterization of equilibrium

• representative agent

• existence of equilibrium

Ref: [KS] Sections 4.1-4.5.

An example of computations can be found here

Suggested exercises on this topic: Exercise 8?

• Example/exercise related to the topic

Suggested exercises on this topic: Exercise 7

Introduction to some elements of equilibrium theory:

• interacting agents in the market

• assumption on endowment

• single agent optimization problem, admissible portfolio, budget constraints

Ref:[KS] Chapter 4.?

• Analysis of the constrains and description of the associated family of unconstrained markets

• Relationship between the optimal performance in each of these markets and the optimal performance in the original market

• Sufficient condition for the existence of a solution to the constrained optimization problem

Ref: [KS] Section 6.1, 6.2, 6.3?

• completion of the proof of the characterization of the upperhedging price

• special case of constant coefficients

Ref: [KS] Section 5.6, 5.7

Exercises in class: corrections/comments from the previous round?

• characterization of the upperhedging price

Ref: [KS] Section 5.6

Some comments about inequalities are here: Note

Some suggested exercises for 2.11 on the topic are here: Exercise 5 and Exercise 6 ?

• admissible consumption/investments schemes and budget constraints

• consistency of the description of the concepts in the market with constraints with the market without constraints: replicability results, unconstrained hedging price.

• upperhedging price and attainability in presence of constraint

• revision of the pricing in an unconstrained market

Introduction to the family of auxiliary markets as a way to introduce the constraint on portfolios into the markets' description: use of the support function

Ref: [KS] Section 5.2 , 5.4, 5.4, 5.5?

• solution of the optimization problem (OP)

• association to the auxiliary optimization problem (aOP)

• Lagrange method/martingale method for the solution via a family of non-constrained problems (uOP)

Ref in [KS] Sections 3.5, 3.6, 3.7

Some exercises suggested for next time are here:Exercise 1 , Exercise 2 , Exercise 3 , and Exercise 4

Introduction of the topic: Pricing in markets with constraint on portfolios.

Ref: [KS] Section 5.2?

• Other basic concepts including the "Budget constraint". Introduction to utility functions.

Ref in [KS] Sections 3.3, 3.4?

• Quick introduction/revision of (continuous) local martingales, classical Ito integration, extended Ito integration

Ref: any book in stochastic integration in the general set-up of continuous local martingales, e.g. Karatzas and Shreve "Brownian motion and stochastic calculus", Rogers and Williams "Diffusions, Markov processes and Martingales" Vol II.?

• Some information on the course

• Detailed description of the background knowledge in stochastic analysis useful for the course (Brownian motion, Ito stochastic integration, Ito formula, SDEs, martingales, local martingales, Girsanvo theorem, Ito representation theorem).

• Beginning of the program: Single-agent consumption and investment. First definitions and concepts.

Ref in: [KS] Sections 3.2, 3.3?