  This memoir presents a systematic study of utility maximization problems for
an investor in constrained and unbounded financial markets. Building upon the
foundational work of Hu et al. (2005) [Ann. Appl. Probab., 15, 1691--1712] in a
bounded framework, we extend our analysis to more challenging unbounded cases.
Our methodology combines quadratic backward stochastic differential equations
with unbounded solutions and convex duality methods. Central to our approach is
the verification of the finite entropy condition, which plays a pivotal role in
solving the underlying utility maximization problems and establishing the
martingale property and convex duality representation of the value processes.
Through four distinct applications, we first study utility indifference
valuation of financial derivatives with unbounded payoffs, uncovering novel
asymptotic behavior as the risk aversion parameter approaches zero or infinity.
Furthermore, we study the regime switching market model with unbounded random
endowments and consumption-investment problems with unbounded random
endowments, both constrained to portfolios chosen from a convex and closed set.
Finally, we investigate investment-consumption problems involving an investor
with Epstein-Zin recursive utility in an unbounded financial market.
