General Stochastic Differential Equations and Control Problems

This book studies stochastic differential equations and stochastic control problems driven by general continuous local martingales with spatial parameters. In this setting, stochastic dynamics are characterized through local characteristics and quadratic variation, allowing the modeling of systems beyond the classical Brownian framework.The book first investigates weak and strong solution concepts for stochastic differential equations driven by continuous local martingales. Weak existence results are established using Euler-type approximation schemes combined with the martingale problem formulation. Strong convergence properties and optimal rates of convergence are also analyzed, providing a rigorous theoretical foundation for stochastic systems driven by general martingale noise.The final part is devoted to stochastic control problems in this general martingale framework, including optimal feedback and singular control problems. Necessary optimality conditions are derived using variational arguments and adjoint equations involving backward stochastic differential equations driven by continuous local martingales.