  This investigation is dedicated to a two-player zero-sum stochastic
differential game (SDG), where a cost function is characterized by a backward
stochastic differential equation (BSDE) with a continuous and monotonic
generator regarding the first unknown variable, which possesses immense
applicability in financial engineering. A verification theorem by virtue of
classical solution of derived Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation is
given. The dynamic programming principle (DPP) and unique weak (viscosity)
solvability of HJBI equation are formulated through comparison theorem for
BSDEs with monotonic generators and stability of viscosity solution. Some new
regularity properties of value function are presented. Finally, we propose
three concrete examples, which are concerned with resp., classical, and
viscosity solution of HJBI equation, as well as a financial application where
an investor with a non-Lipschitzian Epstein-Zin utility deals with market
friction to maximize her utility preference.
