Recursive stochastic differential games with non-Lipschitzian generators and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equation

Abstract: 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.