In this thesis, we will be investigating the problem of maximizing intergenerational equity in the consumption of a naturally exhaustible resource, taking into account both resource stock dynamics and environmental quality. Our model will be governed by two coupled stochastic differential equations (SDEs): one describing the evolution of the resource stock and the other the dynamics of environmental quality. Our objective will be based on optimizing consumption while balancing the trade-offs between resource depletion and environmental degradation, with an exclusive focus on fairness across generations. Specifically, our objective function will incorporate penalized consumption and social costs to reflect the need to curtail excessive consumption and mitigate environmental impacts.