Description of the lecture series (tentative plan)
Detailed plan of 10 lectures on the topic of Convex Duality Methods in Mathematical Finance is as follows:
1) The market model. We describe a general market model and the preferences of the agents trading in the market. We focus on incompleteness of the market and motivate the need of an Orlicz space approach.

2) The fundamental theorem of asset pricing. We address the classical topic on No Arbitrage and existence of equivalent local/sigma martingale measures. We introduce the notion of No Market Free Lunch that depends on the preferences of the agents and compare it with the classical notions of No Arbitrage and NFLVR.

3) On the super-replication value. We investigate the super-replication cost in incomplete markets under the no arbitrage assumption and develop a super-replication value based on the utility of the agent.

4) Utility maximization from terminal wealth. When the price processes follow a (possibly unbounded) semimartingale it is convenient to embed utility maximization in the Orlicz space naturally induced by the utility function. By applying the duality approach we show
existence of the optimal solution. We recover classical results and show that in this general framework a singular component in the dual variables may arise.

5) On the supermartingale property of the optimal wealth process. The optimal wealth process is a supermartingale under each semimartingale measure with finite generalized entropy. This result is proved by analyzing the dynamic version of the dual problem.

6) Utility maximization with a random endowment. We address the same topic of lecture 4 but we allow for a random endowment. We show that under very general assumptions on the claim we still can prove the duality relation and the existence of the optimal solution. We study the indifference price of a claim and show that in general context the indifference price is a risk measure on the Orlicz space associated to the utility function.

7) Risk measures. We study the dual representation of risk measures defined on Frechet lattices and in particular on Orlicz spaces. We emphasize that the order lower semicontinuity property (the Fatou property) is sufficient for the representation of the risk measure in terms of order continuous linear functional.

8) Topics in the theory of risk measures. Law invariant risk measures. Cash sub-additive risk measures. Pareto optimal allocations.

9) Topics in the theory of risk measures. The dynamics of a risk measure. Risk measures and g-expectations. Risk measures for processes.

10) Some open problems.

Statistics & Applied Probability
University of California
Santa Barbara, California 93106-3110
(805) 893-2129