Theorems of comparison and stability with probability 1 for one-dimensional stochastic differential equations (Q511326)

The authors study the following scalar equations with symmetric integral \[ \begin{aligned} & d \xi_t^{(1)} = \sigma^{(1)} (t,\xi_t^{(1)}) * dX(t) + b^{(1)}(t,\xi_t^{(1)}) dt,\\ & \xi_t^{(1)}|_{t=t_0} = \xi_0^{(1)}, \\ & d \xi_t^{(2)} = \sigma^{(2)} (t,\xi_t^{(2)}) * dX(t) + b^{(2)}(t,\xi_t^{(2)}) dt,\\ & \xi_t^{(2)}|_{t=t_0} = \xi_0^{(2)}.\end{aligned} \] Their aim is to prove comparison theorems in the case where \(\sigma^{(1)} \neq \sigma^{(2)}\).