Generalized integration and stochastic ODEs

Stochastic forward integrals for processes more general than semimartingales are shown to exist, generalized forms of Itô-Wentzell formula and covariation formula are proved, and one-dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved. This generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift.