# 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.

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.

MSC Code:

91G30

Keywords: