I develop a macroeconomic model accounting for global warming using stochastic control to study how climate policies can maximize welfare through incentivizing green capital investment. Mathematically, this system defines a mean-field game, which I represent using nonlinear PDEs resembling equations of motion for classical particles in a physical system. However, this economic system is much harder to solve due to constraints of dynamic optimization and belief consistency. I first rigorously develop stochastic control theory and draw extensive parallels to Lagrangian and Hamiltonian mechanics. I then consider a continuum of economic agents who optimize investment decisions between “green” and “brown” capital types, the latter of which drives increases in a stochastic temperature process that damages capital productivity. I leverage equation-informed neural networks to solve for agent value functions and the evolution of the distributions of economic variables to quantify welfare and climate-transition trajectories under a central-planner economy and a decentralized equilibrium under multiple climate policies. The technical contributions of this thesis include obtaining global solutions for decentralized equilibrium using deep learning, for which I achieve upper bounds of 2 ×10−4 mean- squared error equation loss for the agent value function, close to the median threshold of 1 ×10−4 for reported convergence in three papers inspiring this work; and validating a deep-learning solution of the central-planner economy with 14% mean relative error versus a finite-difference solution and 4% normalized difference from an analytic boundary value. The economic-policy-relevant contributions include quantifying a 2% normalized reduction in agent value from the central-planner economy to the decentralized economy without climate policy and an 8% normalized reduction in agent value from an optimal constant carbon tax to a stochastic tax representing political turnover. My results further suggest that to mitigate weaker incentives under a stochastic carbon tax, a welfare-maximizing government should invest carbon-tax revenue directly into green capital rather than transferring monetary revenue back to the population.