  This paper studies a competitive optimal portfolio selection problem in a
model where the interest rate, the appreciation rate and volatility rate of the
risky asset are all stochastic processes, thus forming a non-Markovian
financial market. In our model, all investors (or agents) aim to obtain an
above-average wealth at the end of the common investment horizon. This
competitive optimal portfolio problem is indeed a non-zero stochastic
differential game problem. The quadratic BSDE theory is applied to tackle the
problem and Nash equilibria in suitable spaces are found. We discuss both the
CARA and CRRA utility cases. For the CARA utility case, there are three
possible scenarios depending on market and competition parameters: a unique
Nash equilibrium, no Nash equilibrium, and infinite Nash equilibria. The Nash
equilibrium is given by the solutions of a quadratic BSDE and a linear BSDE
with unbounded coefficient when it is unique. Different from the
wealth-independent Nash equilibria in the existing literature, the equilibrium
in our paper is of feedback form of wealth. For the CRRA utility case, the
issue is a bit more complicated than the CARA utility case. We prove the
solvability of a new kind of quadratic BSDEs with unbounded coefficients. A
decoupling technology is used to relate the Nash equilibrium to a series of
1-dimensional quadratic BSDEs. With the help of this decoupling technology, we
can even give the limiting strategies for both cases when the number of agent
tends to be infinite.
