  We study the problem of optimal long term portfolio selection with a view to
beat a benchmark. Two kinds of objectives are considered. One concerns the
probability of outperforming the benchmark and seeks either to minimise the
decay rate of the probability that the portfolio exceeds the benchmark or to
maximise the decay rate that the portfolio falls short. The other criterion
concerns the growth rate of the risk-sensitive utility of wealth which has to
be either minimised, for the risk-averse investor, or maximised, for the
risk-seeking investor. It is assumed that the mean returns and volatilities of
the securities are affected by an economic factor, possibly, in a nonlinear
fashion. The economic factor and the benchmark are modelled with general It\^o
differential equations. The results identify optimal portfolios and produce
decay, or growth, rates. The portfolios have the form of time-homogeneous
functions of the economic factor. Furthermore, a uniform treatment is given to
the out- and under- performance probability optimisation as well as to the
risk-averse and risk-seeking optimisation. In particular, it is shown that
there exists a portfolio that optimises the decay rates of both the
outperfomance probability and the underperformance probability. Whereas earlier
research on the subject has relied on the techniques of stochastic optimal
control and dynamic programming, in this contribution the quantities of
interest are studied directly by employing the methods of the large deviation
theory. The key to the analysis is to recognise the setup in question as a case
of coupled diffusions with time scale separation, with the economic factor
representing "the fast process".
