  We consider the forward investment problem in market models where the stock
prices are continuous semimartingales adapted to a Brownian filtration. We
construct a broad class of forward performance processes with initial
conditions of power mixture type, $u(x) = \int_{\mathbb{I}}
\frac{x^{1-\gamma}}{1-\gamma }\nu(\mathrm{d} \gamma)$. We proceed to define and
fully characterize two-power mixture forward performance processes with
constant risk aversion coefficients in the interval $(0,1)$, and derive
properties of two-power mixture forward performance processes when the risk
aversion coefficients are continuous stochastic processes. Finally, we discuss
the problem of managing an investment pool of two investors, whose respective
preferences evolve as power forward performance processes.
