  In an Ito-diffusion market, two fund managers trade under relative
performance concerns. For both the asset specialization and diversification
settings, we analyze the passive and competitive cases. We measure the
performance of the managers' strategies via forward relative performance
criteria, leading to the respective notions of forward best-response criterion
and forward Nash equilibrium. The motivation to develop such criteria comes
from the need to relax various crucial, but quite stringent, existing
assumptions -- such as, the a priori choices of both the market model and the
investment horizon, the commonality of the latter for both managers as well as
the full a priori knowledge of the competitor's policies for the best-response
case. We focus on locally riskless criteria and deduce the random forward
equations. We solve the CRRA cases, thus also extending the related results in
the classical setting. An important by-product of the work herein is the
development of forward performance criteria for investment problems in
Ito-diffusion markets under the presence of correlated random endowment process
for both the perfectly and the incomplete market cases.
