I have system of three SDE's. They have the following form:
$$dx=x(zF_1(x,z)-d)dt+\sigma_1 dW_1(t)$$ $$dy=y(zF_2(x,z)-d)dt+\sigma_2 dW(t)_2$$ $$dz=((x+y)d-z(xF_1(x,z)+yF_2(x,z)))dt+\sigma_3 dW_3(t)$$
Numerical integration(Euler-Maruyama ) seems to suggest that the fraction $\frac{x}{x+y}$ is increasing with respect to $\sigma_3$. I understand that these are highly non linear equations, but I was wondering if any body has has any ideas about how to approach the problem of the asymptotic behavior of it, especially in the case that $\lim\sigma_1,\sigma_2\to 0$
Also, x+y+z==1 by construction