We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem.

翻译：本文致力于构建无偏估计量，以逼近McKean-Vlasov随机微分方程（MVSDE）的平稳分布。此类过程是随机分析中的重要分支，在数理金融、生物学与观点动力学等应用领域频繁出现。通常其平稳分布未知，且无法对这类过程进行精确模拟。因此，研究者常需借助时间离散化方案，但这会引入离散化误差以及因无法模拟真实平稳分布而产生的偏差。为消除此类偏差，受无偏蒙特卡洛文献启发，我们提出了一种新型无偏估计量。在特定假设条件下，我们证明了该估计量的无偏性。为此，我们需通过恰当的离散化方案，建立各类离散时间过程遍历其不变测度的理论结果。本文通过一系列MVSDE的数值实验（包括Currie-Weiss模型、三维神经科学模型及参数估计问题）验证了所提无偏估计量的有效性。