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Transition probabilities for stochastic differential equations using the Laplace approximation: Analysis of the continuous-time limit

Uffe Høgsbro Thygesen

TL;DR

This work develops a continuous-time Laplace approximation for transition densities of stochastic differential equations by centering the path integral around the most probable trajectory via a minimum-energy control problem and a Girsanov shift. It expresses the resulting approximation through a sequence of ODEs: the canonical equations for the most probable path, a Riccati equation for the Hessian of the value function, and a Lyapunov equation for the variance of fluctuations, enabling efficient computation of $p(0,x_0,T,x_T)$. The analysis clarifies the separation of discretization error from Laplace error, demonstrates exactness for geometric Brownian motion, and reveals limitations when far-path contributions matter (e.g., double-well). The numerical experiments show Strang splitting improves discretization order to 2, and provide practical guidance on step-sizes and accuracy scaling with time horizon and noise level, offering a framework to build higher-order SDE discretizations for time-series inference.

Abstract

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.

Transition probabilities for stochastic differential equations using the Laplace approximation: Analysis of the continuous-time limit

TL;DR

This work develops a continuous-time Laplace approximation for transition densities of stochastic differential equations by centering the path integral around the most probable trajectory via a minimum-energy control problem and a Girsanov shift. It expresses the resulting approximation through a sequence of ODEs: the canonical equations for the most probable path, a Riccati equation for the Hessian of the value function, and a Lyapunov equation for the variance of fluctuations, enabling efficient computation of . The analysis clarifies the separation of discretization error from Laplace error, demonstrates exactness for geometric Brownian motion, and reveals limitations when far-path contributions matter (e.g., double-well). The numerical experiments show Strang splitting improves discretization order to 2, and provide practical guidance on step-sizes and accuracy scaling with time horizon and noise level, offering a framework to build higher-order SDE discretizations for time-series inference.

Abstract

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.

Paper Structure

This paper contains 14 sections, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Transition densities and their discretized Laplace approximation
  3. The continuous-time limit
  4. The most probable path
  5. Centering the probability at the optimal path
  6. Laplace vs. weak noise approximations
  7. Small noise expansion
  8. Examples
  9. Geometric Brownian motion
  10. The double well
  11. Order analysis for discretization schemes
  12. A Strang splitting algorithm
  13. The accuracy of the Laplace approximation
  14. Discussion

Figures (2)

  • Figure 1: Errors in computing the transition densities for the Cox-Ingersoll-Ross process: Comparison of the discrete-time Laplace approximation $\hat{p}(\cdot,h)$ for different time steps $h$ and for the two discretization methods , the limiting continuous-time Laplace approximation $\hat{p}(\cdot,0)$, and the true transition density $p(\cdot)$. See text for full description.
  • Figure 2: Errors in the continuous-time Laplace approximation for the Cox-Ingersoll-Ross process \ref{['eq:CIR']}: Comparison of the theoretical transition densities with the expression \ref{['eq:final-result']}. Left panel: The effect of varying the noise intensity $\gamma$. Right panel: The effect of varying the final time $T$. Base parameters are $\lambda=1$, $\xi=1$, $\gamma=0.5$, $T=1$, $x_0=0.75$. See the text for how the terminal point $x_T$ is chosen.