Dynamic Utilities and Long Term Interest Rates

Saturday, May 19, 2012 - 2:15pm - 3:00pm
Keller 3-180
Nicole El Karoui (École Polytechnique)
A large debate is open for several years within mathematical nance
about the criterion to optimise, in particular for long term policy. From
the perspective of public decision, such strategy must be time consistent.
Moreover the use of adaptative criterion is necessary to integrate some major
variation in the environment. A typical example is the forward utilities
introduced by M. Musiela and T. Zariphopoulou in 2003, for which there is
no-prespecied trading horizon.
First we characterize utility random elds by showing that the associated
marginal utility is a monotonic solution of SDE with random coecients;
its inverse, the marginal conjuguate utility is solution of a SPDE driven
by the adjoint elliptic operator. When forward utilities satisfy a property
of consistency with a given incomplete nancial market, as in the classical
case, dynamic utilities and its conjugate may be characterized in terms of
Hamilton Jacobi Bellman SPEs as value functions of control problems. More
interesting is the splitting property of the marginal utility in terms of optimal
processes, leading to an explicit solution given by the composition of the
optimal conjugate process with the inverse of the optimal wealth.
Then, it is possible to generate time consistency yield curves by indif-
ference pricing. In the controversy on the discount rate used in nancing
long term projects, such a criterion leads to a time consistency yield curve
depending of the wealth of the economy.
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