I am new to stochastic differential equations, and I don’t understand the formalism of g and dW. I need to solve the differential equation system with the noise which enters in multiple equations and then extract some averages from the solution.

For example, lets consider an example system with a noise \xi(t): \begin{cases} \dot x = 1 - \xi(t) \\ \dot y = x + \xi(t) \end{cases}
where the noise is white \langle \xi(t) \xi(t') \rangle = \delta(t-t'). How can I formulate the system for DifferentialEquations.jl?

From the solution, I need to extract averages \langle x \rangle (t) and \langle y \rangle (t) as well as average of some arbitrary function \langle f(x,y) \rangle (t). Additionally, I would need an average of functional \langle F([x(t)],[y(t)] ) \rangle. Is it possible to do thoose things with DifferentialEquations.jl?