We recently proposed a method for estimation of states and parameters in stochastic differential equations, which included intermediate time points between observations and used the Laplace approximation to integrate out these intermediate states. In this paper, we establish a Laplace approximation for the transition probabilities in the continuous-time limit where the computational time step between intermediate states vanishes. Our technique views the driving Brownian motion as a control, casts the problem as one of minimum effort control between two states, and employs a Girsanov shift of probability measure as well as a weak noise approximation to obtain the Laplace approximation. We demonstrate the technique with examples; one where the approximation is exact due to a property of coordinate transforms, and one where contributions from non-near paths impair the approximation. We assess the order of discrete-time scheme, and demonstrate the Strang splitting leads to higher order and accuracy than Euler-type discretization. Finally, we investigate numerically how the accuracy of the approximation depends on the noise intensity and the length of the time interval.

翻译：我们近期提出了一种用于估计随机微分方程中状态与参数的方法，该方法通过在观测点之间插入中间时间点，并利用拉普拉斯近似对这些中间状态进行积分。本文针对中间状态间计算时间步长趋于零的连续时间极限情形，建立了转移概率的拉普拉斯近似方法。我们的技术将驱动布朗运动视为控制变量，将问题转化为两点间最小能耗控制问题，并通过Girsanov测度变换结合弱噪声近似获得拉普拉斯近似。我们通过算例验证该技术：在第一个算例中，由于坐标变换的特性，该近似具有精确性；而在第二个算例中，非邻近路径的贡献会削弱近似效果。我们评估了离散时间格式的阶数，并证明Strang分裂法相比欧拉型离散化具有更高阶精度。最后，我们通过数值实验研究了近似精度对噪声强度与时间区间长度的依赖关系。