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 {$\mathcal{O}(\epsilon^{-\tfrac{4}{2H+1}})$, where $H$ is the Hurst parameter. If one uses a single level MCMC method then the cost is $\mathcal{O}(\epsilon^{-\tfrac{2(2H+3)}{2H+1}})$} 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）的贝叶斯参数推断问题。SVE广泛应用于物理学和数理金融等领域，在后者中可用于表征未观测波动率过程的长期记忆特性。在许多实际场景中，需对SVE进行时间离散化处理，并基于离散化过程对应的后验分布进行参数推断。参照近期SVE时间离散化研究（如Richard等，2021），本文采用Euler-Maruyama方法进行前述离散化，进而展示多层马尔可夫链蒙特卡洛方法（Jasra等，2018）在此框架下的应用。通过实例分析，我们给出证明表明：实现均方误差（MSE）为$\mathcal{O}(\epsilon^2)$（$\epsilon>0$）的计算代价为$\mathcal{O}(\epsilon^{-\tfrac{4}{2H+1}})$，其中$H$为赫斯特参数。若采用单层MCMC方法，达成相同MSE的计算代价则为$\mathcal{O}(\epsilon^{-\tfrac{2(2H+3)}{2H+1}})$。我们以状态空间模型和随机波动率模型为例验证上述结论，并基于真实数据对后者进行了实证分析。