Persistence in probability for stochastic delay equations with multiplicative noise
Abstract
We study the stochastic delay equation dX(t)=[−γX(t)+r f(X(t−τ))] dt+σX(t) dW(t), where γ,r,τ∈(0,∞), σ∈[0,∞), and with f either globally Lipschitz or globally bounded and locally Lipschitz, with f(0)=0 and f(x)≥0 for all x>0. Under the condition rliminfx→ 0+f(x)/x>γ+1/2σ2, independent of the time delay τ, 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.
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BibTeX
@misc{bosch2026persistence,
title = {{Persistence in probability for stochastic delay equations with multiplicative noise}},
author = {Mark van den Bosch and John Ioannis Stavroulakis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21599},
eprint = {2610.21599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21599},
}
RIS
TY - UNPB
TI - Persistence in probability for stochastic delay equations with multiplicative noise
AU - Mark van den Bosch
AU - John Ioannis Stavroulakis
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21599
UR - https://arxiv.org/abs/2610.21599
AB - We study the stochastic delay equation dX(t)=[−γX(t)+r f(X(t−τ))] dt+σX(t) dW(t), where γ,r,τ∈(0,∞), σ∈[0,∞), and with f either globally Lipschitz or globally bounded and locally Lipschitz, with f(0)=0 and f(x)≥0 for all x>0. Under the condition rliminf_x→ 0^+f(x)/x>γ+1/2σ^2, independent of the time delay τ, 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.
ER -