  We develop an approach for two player constraint zero-sum and nonzero-sum
stochastic differential games, which are modeled by Markov regime-switching
jump-diffusion processes. We provide the relations between a usual stochastic
optimal control setting and a Lagrangian method. In this context, we prove
corresponding theorems for two different type of constraints, which lead us to
find real valued and stochastic Lagrange multipliers, respectively. Then, we
illustrate our results for a nonzero-sum game problem with stochastic maximum
principle technique. Our application is an example of cooperation between a
bank and an insurance company, which is a popular, well-known business
agreement type, called Bancassurance.
