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：我们致力于开发无偏估计量，以逼近 McKean-Vlasov 随机微分方程（MVSDEs）的平稳分布。这是一类重要的随机过程，常见于数学金融、生物学和观点动力学等应用领域。通常，其平稳分布是未知的，且无法精确模拟此类过程。因此，通常需要采用时间离散化方案，但这会引入离散化偏差以及因无法模拟相应平稳分布而产生的偏差。为克服此偏差，受无偏蒙特卡洛文献的启发，我们提出了一种新的无偏估计量。我们在一定假设条件下证明了该估计量的无偏性。为此，我们需要通过适当的离散化方案，研究各类离散时间过程遍历其不变测度的遍历性结果。我们在一系列 MVSDEs 上进行了大量数值实验，以验证所提无偏估计量的有效性。这些示例包括 Currie-Weiss 模型、一个三维神经科学模型以及一个参数估计问题。