Orateur
Description
In the real world, decision making under uncertainty is often viewed as an opti-
mization problem under choice criterium, and most of theory focuses on the deriva-
tion of the "optimal decision" and its out-comes. But, poor information is available
on the criterium yielding to these observed data. The interesting problem to infer
the unknown criterium from the known results is an example of inverse problem.
Here we are concerned with a very simple version of the problem: what does ob-
servation of the "optimal" out-put tell us about the preference, expressed in terms
of expected utility; in Economics, this question was pioneered by the american
economist Samuelson in1938.
Typically we try to reproduce the properties of the stochastic value function of
a portfolio optimization problem in finance, which satisfies the first order condi-
tion U (t, z)). In particular, the utility process U is a strictly concave stochastic
family, parametrized by a number z ∈ R + (z 7→ U (t, z)), and the characteris-
tic process X c = (X t c (x)) is a non negative monotonic process with respect to
its initial condition x, satisfying the martingale condition U (t, X t c (x)) is a martin-
gale, with initial condition U (0, x) = u(x). We first introduce the adjoint process
Y t (u x (x)) = U x (t, X t c (x)) which is a characteristic process for the Fenchel transform
of U if and only if X t c (x)Y t (u x (x)) is a martingale. The minimal property is the
martingale property of Y t (u x (x)) with the x-derivative of X t c (x), which is sufficient
to reconstruct U from U x (t, x) = Y t (u x ((X t c ) −1 (x))). Obviously, in general, with-
out additional constraints, the characterization is not unique. Various example are
given, in general motivated by finance or economics: contraints on a characteristic
portfolio in a economy at the equilibrium, thoptimal portfolio for a in complete
financial market, under strongly orthogonality between X and Y , the mixture of
different economies....In any case, the results hold for general but monotonic pro-
cesses, without semimartingale assumptions.
