Consumption/investment problem when the investment opportunity set can be enlarged by information gathering

for x > 0. The maximizing (c = C(x), π= π(x)) satisfies V ′(x) = c−γ, π = − V ′(x) V ′′(x) (α− r1m)Σ = C(x) γC ′(x) (α− r1m)Σ, if V ′′ < 0, which will be checked later. Using an idea from Karatzas et al. [2], the Bellman equation (A.1) can be linearized by introducing a function X(c) := C−1(x). That is, the Bellman equation can be transformed into the the following equation κX ′′(c) = − − β − κ) + κ c X ′(c) + γ c2 (rX(c)− c), c > 0,