In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest, SVEs must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. Based upon recent studies on time-discretization of SVEs (e.g. Richard et al. 2021), we use Euler-Maruyama methods for the afore-mentioned discretization. We then show how multilevel Markov chain Monte Carlo (MCMC) methods (Jasra et al. 2018) can be applied in this context. In the examples we study, we give a proof that shows that the cost to achieve a mean square error (MSE) of $\mathcal{O}(\epsilon^2)$, $\epsilon>0$, is $\mathcal{O}(\epsilon^{-20/9})$. If one uses a single level MCMC method then the cost is $\mathcal{O}(\epsilon^{-38/9})$ to achieve the same MSE. We illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real data.

翻译：本文研究一类部分观测随机Volterra方程（SVE）的贝叶斯参数推断。SVE广泛应用于物理和数学金融等领域，在后一领域中，可用于表示未观测波动率过程中的长记忆性。在许多实际场景中，必须对SVE进行时间离散化处理，并基于与该离散化过程相关的后验分布进行参数推断。基于近期关于SVE时间离散化方法的研究（如Richard等，2021），我们采用Euler-Maruyama方法进行前述离散化，并进一步展示了如何在相关背景下应用多层马尔可夫链蒙特卡洛（MCMC）方法（Jasra等，2018）。在案例研究中，我们证明了：为实现均方误差（MSE）为$\mathcal{O}(\epsilon^2)$（$\epsilon>0$）的估计，计算成本为$\mathcal{O}(\epsilon^{-20/9})$；而采用单层MCMC方法时，实现相同MSE所需的计算成本为$\mathcal{O}(\epsilon^{-38/9})$。我们分别在状态空间模型和随机波动率模型中对结果进行了验证，并将后者应用于实际数据。