Before revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting  .
After revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes, and prove that the leading order of the error process scales with the square of the time discretization , making it the first second order method in this setting . Examples are provided demonstrating both the main analytical results, and the reduction in computational complexity achieved with the approximate methods .