Nonzero-Sum Stochastic Differential Game between Controller and Stopper for Jump Diffusions

and Applied Analysis 3 Fix an open solvency set S ⊂ R. Let τS = inf {t > 0; Y (t) ∉ S} (10) be the bankruptcy time. τS is the first time at which the stochastic process Y(t) exits the solvency set S. Similar optimal control problems in which the terminal time is governed by a stopping criterion are considered in [19–21] in the deterministic case. Let f i : Rk × K → R and g i : Rk → R be given functions, for i = 1, 2. Let A be a family of admissible controls, contained in the set of u(⋅) such that (9) has a unique strong solution and