We present a Bayesian synthetic likelihood method to estimate both the parameters and their uncertainty in systems of stochastic differential equations. Together with novel summary statistics the method provides a generic and model-agnostic estimation procedure and is shown to perform well even for small observational data sets and biased observations of latent processes. Moreover, a strategy for assessing the goodness of the model fit to the observational data is provided. The combination of the aforementioned features differentiates our approach from other well-established estimation methods. We would like to stress the fact that the algorithm is pleasingly parallel and thus well suited for implementation on modern computing hardware. We test and compare the method to maximum likelihood, filtering and transition density estimation methods on a number of practically relevant examples from mathematical finance. Additionally, we analyze how to treat the lack-of-fit in situations where the model is biased due to the necessity of using proxies in place of unobserved volatility.
