Persistence in probability for stochastic delay equations with multiplicative noise

We study the stochastic delay equation \begin{equation*} dX(t)=[-γX(t)+r f(X(t-τ))],dt+σX(t),dW(t), \end{equation*} where $\gamma ,r,\tau \in (0,\mathrm{\infty })$, $\sigma \in [0,\mathrm{\infty })$, and with $f$ either globally Lipschitz or globally bounded and locally Lipschitz, with $f(0)=0$ and $f(x)\ge 0$ for all $x>0$. Under the condition \begin{equation*} r\liminf_{x\to 0^{+}}\frac{f(x)}{x}>γ+\frac{1}{2}σ^2, \end{equation*} independent of the time delay $\tau ,$ we obtain that every solution $X$ with nonnegative initial history and $X(0)>0$ is bounded away from zero in probability. This result then confirms a recent conjecture of van den Bosch, van Gaans, and Verduyn Lunel, and yields a non-trivial invariant measure for, e.g., the Mackey–Glass or Nicholson’s blowflies equation with multiplicative noise. To get to this result, we apply a transformation to the solution process, essentially rewriting the equation using a stochastic integrating factor. A comparison with a linear equation gives contraction estimates for negative conditional moments and hence boundedness away from zero in probability. The same argument yields persistence whenever the linear comparison equation has a positive first Lyapunov exponent.