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 types 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 the 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.

In this research article, we study a stochastic control problem in a theoretical frame to solve a constrained task under memory impact. The nature of memory is modeled by Stochastic Differential Delay Equations and our state process evolves according to a jump-diffusion process with time-delay. We work on two specific types of constraints, which are described in the stochastic control problem as running gain components. We develop two theorems for corresponding deterministic and stochastic Lagrange multipliers. Furthermore, these theorems are applicable to a wide range of continuous-time stochastic optimal control problems in a diversified scientific area such as Operations Research, Biology, Computer Science, Engineering and Finance. Here, in this work, we apply our results to a financial application to investigate the optimal consumption process of a company via its wealth process with historical performance. We utilize the stochastic maximum principle, which is one of the main methods of continuous-time Stochastic Optimal Control theory. Moreover, we compute a real-valued Lagrange multiplier and clarify the relation between this value and the specified constraint.

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.