Power mixture forward performance processes

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.