  Within a common arbitrage-free semimartingale financial market we consider
the problem of determining all Nash equilibrium investment strategies for $n$
agents who try to maximize the expected utility of their relative wealth. The
utility function can be rather general here. Exploiting the linearity of the
stochastic integral and making use of the classical pricing theory we are able
to express all Nash equilibrium investment strategies in terms of the optimal
strategies for the classical one agent expected utility problems. The
corresponding mean field problem is solved in the same way. We give four
applications of specific financial markets and compare our results with those
given in the literature.
