16:00
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:
(i) The optimal terminal wealth X*(T) := Xφ* (T) of the classical problem to
maximise the expected U-utility of the terminal wealth Xφ(T) generated by admissible
portfolios φ(t); 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale;
(ii) The optimal scenario dQ*/dP of the dual problem to minimise the expected
V -value of dQ/dP over a family of equivalent local martingale measures Q. Here V is
the convex dual function of the concave function U.
In this talk we consider markets modeled by Itô-Lėvy processes, and we present
in a first part a new proof of the above result in this setting, based on the maximum
principle in stochastic control theory. An advantage with our approach is that it also
gives an explicit relation between the optimal portfolio φ* and the optimal scenario
Q*, in terms of backward stochastic differential equations. In a second part we present
robust (model uncertainty) versions of the optimization problems in (i) and (ii), and
we prove a relation between them. We illustrate the results with explicit examples.
The presentation is based on recent joint work with Bernt ¬Oksendal, University of
Oslo, Norway.