A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: %DIFDELCMD < myenumerate \item %%%  The optimal terminal wealth X^*(T) : = X_{\varphi^*}(T) of the classical problem to maximize the expected U-utility of the terminal wealth X_{\varphi}(T) generated by admissible portfolios \varphi(t) ; 0 \leq t \leq T in a market with the risky asset price process modeled as a semimartingale %DIFDELCMD < \item %%%  The optimal scenario dQ^*{dP} of the dual problem to minimize 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 paper we consider markets modeled by It\^o-L\'evy processes , and in the first part we give 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\emph{ [  an explicit relation between the optimal portfolio \varphi^* and the optimal measure Q^* , in terms of backward stochastic differential equations . In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a  relation between them. In particular, we show explicitly how to get from the solution of one of the problems to the solution of the other. We illustrate the results with explicit examples.
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: %DIFDELCMD < myenumerate \item %%% (i) The optimal terminal wealth X^*(T) : = X_{\varphi^*}(T) of the problem to maximize the expected U-utility of the terminal wealth X_{\varphi}(T) generated by admissible portfolios \varphi(t) , 0 \leq t \leq T in a market with the risky asset price process modeled as a semimartingale %DIFDELCMD < \item %%% ; (ii) The optimal scenario dQ^*{dP} of the dual problem to minimize the expected V-value of dQ{dP} over a family of equivalent local martingale measures Q , where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by It\^o-L\'evy processes . In the first part we use the maximum principle in stochastic control theory to extend the above relation to a\emph{dynamic relation, valid for all t \in[0,T . We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, 0 \leq t \leq T. In the terminal time case t=T we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \varphi^* and the optimal measure Q^* . We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim . In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.
A
(i)
D
classical
R
;
,
A
; (ii)
R
. Here
, where
R
dual
conjugate
R
, and in
. In
R
give a new proof of the above result in this setting, based on the
use the
R
. An advantage with our approach is that it also gives
to extend the above relation to a
A
dynamic
A
relation, valid for all t \in
A
0,T
A
. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, 0 \leq t \leq T. In the terminal time case t=T we recover the classical duality connection above. We get moreover
R
, in terms of backward stochastic differential equations
. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim
A
similar dynamic
D
explicitly
D
the solution of