One-Dimensional Stratonovich Differential Equations

Abstract

We consider one-dimensional stochastic differential equations of the Stratonovich type: $dX_t = \sum_i\sigma_i(t, w, X_t)\circ dZ^i_t + \sum_k h_k(t, w, X_t)dA^k_t,$ where $Z^i$ are continuous semimartingales, and $A^k$ are continuous finite variation processes. We extend the definition of the Fisk-Stratonovich integral for a large class of coefficients $\sigma_i$, and under suitable conditions we prove existence and uniqueness for that equation.