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 superreplication value. We investigate the superreplication
cost in incomplete markets under the no arbitrage assumption and develop
a superreplication 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 subadditive risk measures. Pareto optimal allocations.
9) Topics in the theory of risk measures. The dynamics of a risk measure.
Risk measures and gexpectations. Risk measures for processes.
10)
Some open problems.
