We consider the problem of Bayesian inference for bi-variate data observed in time but with observation times which occur non-synchronously. In particular, this occurs in a wide variety of applications in finance, such as high-frequency trading or crude oil futures trading. We adopt a diffusion model for the data and formulate a Bayesian model with priors on unknown parameters along with a latent representation for the the so-called missing data. We then consider computational methodology to fit the model using Markov chain Monte Carlo (MCMC). We have to resort to time-discretization methods as the complete data likelihood is intractable and this can cause considerable issues for MCMC when the data are observed in low frequencies. In a high frequency observation frequencies we present a simple particle MCMC method based on an Euler--Maruyama time discretization, which can be enhanced using multilevel Monte Carlo (MLMC). In the low frequency observation regime we introduce a novel bridging representation of the posterior in continuous time to deal with the issues of MCMC in this case. This representation is discretized and fitted using MCMC and MLMC. We apply our methodology to real and simulated data to establish the efficacy of our methodology.

翻译：本文研究双变量时间序列数据在非同步观测时间下的贝叶斯推断问题。该问题在金融领域的诸多应用中广泛存在，例如高频交易或原油期货交易。我们采用扩散模型描述数据，构建了包含未知参数先验分布及潜在缺失数据表示的贝叶斯模型。随后探讨基于马尔可夫链蒙特卡洛（MCMC）的模型拟合计算方法。由于完整数据似然函数难以处理，必须采用时间离散化方法，这在数据低频观测场景中会给MCMC带来显著挑战。针对高频观测场景，我们提出基于欧拉-丸山时间离散化的简易粒子MCMC方法，并可通过多级蒙特卡洛（MLMC）进行增强。对于低频观测场景，我们引入连续时间后验分布的新型桥接表示法以解决该情境下的MCMC难题，该方法经离散化后可通过MCMC与MLMC进行拟合。我们通过实际数据与模拟数据验证了所提方法的有效性。