Regular-Impulse Control for Air Pollution Management: A Stochastic Differential Game Approach

The work examines a finite-horizon, two-player non-zero sum stochastic differential game describing a pollution abatement initiative. A mining company (Player 1) controls piecewise continuous resource extraction rate, while the government (Player 2) applies impulse control penalty interventions to compel the company to reduce pollution levels. Both players aim to minimise the expected total discounted operation and intervention costs. The main result is presented as quasi-variational inequalities (QVIs), incorporating analytical characterisation of Nash equilibrium strategies. To illustrate the applicability of theoretical results, a scalar linear-quadratic example is discussed, and numerical experiments are implemented. These demonstrate that the continuation region stretches during the initial period, reaching a peak level, before experiencing smooth contraction as time progresses towards the final stages of the company's operations. The results also demonstrate that there is a positive relationship between the continuation region boundary and the post-intervention reset level. The research findings stress the importance of the dynamic trade-off between the intensity of economic activities and optimal implementation of environmental regulation policies in uncertain contexts, emphasising the nexus between regulatory flexibility and corrective severity.