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We develop an approach for two player constraint zero-sum and nonzero-sumstochastic differential games, which are modeled by Markov regime-switchingjump-diffusion processes. We provide the relations between a usual stochasticoptimal control setting and a Lagrangian method. In this context, we provecorresponding theorems for two different type of constraints, which lead us tofind real valued and stochastic Lagrange multipliers, respectively. Then, weillustrate our results for a nonzero-sum game problem with stochastic maximumprinciple technique. Our application is an example of cooperation between abank and an insurance company, which is a popular, well-known businessagreement type, called Bancassurance.

A stochastic control approach for constrained stochastic differential games with jumps and regimes