  This paper studies finite-time optimal consumption-investment problems with
power, logarithmic and exponential utilities, in a regime switching market with
random coefficients, subject to coupled constraints on the consumption and
investment strategies. We provide explicit optimal consumption-investment
strategies and optimal values for the problems in terms of the solutions to
some diagonally quadratic backward stochastic differential equation (BSDE)
systems and linear BSDE systems with unbound coefficients. Some of these BSDEs
are new in the literature and solving them is one of the main theoretical
contributions of this paper. We accomplish the latter by applying the
truncation, approximation technique to get some a priori uniformly lower and
upper bounds for their solutions.
