Date
2009
Dewey
Probabilités et mathématiques appliquées
Sujet
Quantile Constraint; Dynamic Programming Equation; Stochastic Target Problem
JEL code
C73
Conference name
Istanbul Workshop on Mathematical Finance
Conference date
05-2009
Conference city
Istanbul
Conference country
Turquie
Type
Communication / Conférence
Abstract (EN)
We consider the problem of finding the minimal initial data of a controlled process which guarantees
to reach a controlled target with a given probability of success or, more generally, with a given level of
expected loss. By suitably increasing the state space and the controls, we show that this problem can
be converted into a stochastic target problem, i.e. find the minimal initial data of a controlled process
which guarantees to reach a controlled target with probability one. Unlike the existing literature on
stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we
provide a new derivation of the dynamic programming equation for general stochastic target problems
with unbounded controls, together with the appropriate boundary conditions. These results are applied
to the problem of quantile hedging in financial mathematics, and are shown to recover the explicit
solution of Follmer and Leukert. We then consider the problem of miximizing a utility function under
this type of quantile constraint. The previous study allows to characterize the domain in which the
value fuction lies and we provide an Hamilton-Jacobi-Bellman representation of the associated value
function. Contrary to standard state constraint problems, the domain is not given a-priori and we do
not need to impose conditions on its boundary.