Consider the delayed stochastic differential equation as below:
$$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$
$$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-Y_{(t-\theta)^+}^\theta)dt +Y_{t}^\theta(1-Y_{t}^\theta)dW_t ,\quad \forall t>0,$$
where $x^+:=\max(x,0)$, $(W_t)_{t\ge 0}$ is a standard Brownian motion and $\theta\ge 0$ is a fixed constant. For any $z\in [0,1]$, the above equations are well posed with $X_0^\theta:=z$ and $Y_0^\theta:=z$. My question is whether we always have
$$\lim_{t\to\infty}X_t^\theta\in \{0,1\},\quad \lim_{t\to\infty}Y_t^\theta\in \{0,1\}?$$
It follows from Almost sure stability of a scalar, nonautonomous, nonlinear SDE that the claim holds for $\theta=0$. I wish to know how the delay $\theta$ may change fundamentally the asymptotic behaviour.
