A wide variety of phenomena of engineering and scientific interest are of a continuous-time nature and can be modeled by stochastic differential equations (SDEs), which represent the evolution of the uncertainty in the states of a system. For systems of this class, some parameters of the SDE might be unknown and the measured data often includes noise, so state and parameter estimators are needed to perform inference and further analysis using the system state path. The distributions of SDEs which are nonlinear or subject to non-Gaussian measurement noise do not admit tractable analytic expressions, so state and parameter estimators for these systems are often approximations based on heuristics, such as the extended and unscented Kalman smoothers, or the prediction error method using nonlinear Kalman filters. However, the Onsager Machlup functional can be used to obtain fictitious densities for the parameters and state-paths of SDEs with analytic expressions. In this thesis, we provide a unified theoretical framework for maximum a posteriori (MAP) estimation of general random variables, possibly infinite-dimensional, and show how the Onsager--Machlup functional can be used to construct the joint MAP state-path and parameter estimator for SDEs. We also prove that the minimum energy estimator, which is often thought to be the MAP state-path estimator, actually gives the state paths associated to the MAP noise paths. Furthermore, we prove that the discretized MAP state-path and parameter estimators, which have emerged recently as powerful alternatives to nonlinear Kalman smoothers, converge hypographically as the discretization step vanishes. Their hypographical limit, however, is the MAP estimator for SDEs when the trapezoidal discretization is used and the minimum energy estimator when the Euler discretization is used, associating different interpretations to each discretized estimate.