Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Simulation of elliptic and hypo-elliptic conditional diffusions

Joris Bierkens, Frank van der Meulen, Moritz Schauer

TL;DR

This work develops a unified framework for simulating diffusion bridges conditioned on linear observations $V=L X_T$ that applies to both uniformly elliptic and hypo-elliptic diffusions, including when $L\neq I$. By constructing density-based guiding terms through a tractable linear auxiliary process $\tilde{X}$ and densities $\rho,\tilde{\rho}$, the authors define guided proposals with drift $b^\circ=b+a\tilde{r}$, and establish existence, endpoint behavior, and absolute continuity results that justify Metropolis–Hastings sampling of diffusion bridges. Key contributions include (i) extending guided proposals to hypo-elliptic and partial observation settings, (ii) rigorous rates at which the guided proposal approaches the conditioned process near $T$ via a scaling $\Delta(t)$, and (iii) explicit Radon–Nikodym derivatives ensuring valid sampling without requiring intractable endpoint densities. The paper also provides tractable hypo-elliptic models and numerical demonstrations on challenging problems such as partially observed integrated and FitzHugh–Nagumo-type diffusions, illustrating practical applicability to parameter estimation and filtering in complex diffusion models.

Abstract

Suppose $X$ is a multidimensional diffusion process. Assume that at time zero the state of $X$ is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix $L$. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in Schauer et al. (2017), can be used in a unified way for both uniformly and hypo-elliptic diffusions, also when $L$ is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.

Simulation of elliptic and hypo-elliptic conditional diffusions

TL;DR

This work develops a unified framework for simulating diffusion bridges conditioned on linear observations that applies to both uniformly elliptic and hypo-elliptic diffusions, including when . By constructing density-based guiding terms through a tractable linear auxiliary process and densities , the authors define guided proposals with drift , and establish existence, endpoint behavior, and absolute continuity results that justify Metropolis–Hastings sampling of diffusion bridges. Key contributions include (i) extending guided proposals to hypo-elliptic and partial observation settings, (ii) rigorous rates at which the guided proposal approaches the conditioned process near via a scaling , and (iii) explicit Radon–Nikodym derivatives ensuring valid sampling without requiring intractable endpoint densities. The paper also provides tractable hypo-elliptic models and numerical demonstrations on challenging problems such as partially observed integrated and FitzHugh–Nagumo-type diffusions, illustrating practical applicability to parameter estimation and filtering in complex diffusion models.

Abstract

Suppose is a multidimensional diffusion process. Assume that at time zero the state of is fully observed, but at time only linear combinations of its components are observed. That is, one only observes the vector for a given matrix . In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in Schauer et al. (2017), can be used in a unified way for both uniformly and hypo-elliptic diffusions, also when is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.

Paper Structure

This paper contains 35 sections, 18 theorems, 170 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Review of DelyonHu and Schauer
  4. Approach
  5. Extending DelyonHu
  6. Extending Schauer
  7. A toy problem
  8. Outline
  9. Frequently used notation
  10. Inequalities
  11. Linear algebra
  12. Stochastic processes
  13. Main results
  14. Existence of guided proposals
  15. Behaviour of guided proposals near the endpoint
  16. ...and 20 more sections

Key Result

Proposition 2.3

Suppose that the matrix valued function $t,x \mapsto \underline{\sigma}$ in the hypo-elliptic model given by eq:descr-hyp1 and eq:descr-hyp2 has rank $k'$ for all $(t,x)$. Furthermore suppose that $\underline \sigma$ and $\underline \beta$ are infinitely often differentiable with respect to $(t,x)$.

Figures (11)

  • Figure 1: Sampled guided diffusion bridges when conditioning on $X_T=1/321/41^{\prime}$ in Example \ref{['numex:nclar3']}.
  • Figure 2: Sampled guided diffusion bridges when conditioning on $L X_T=1/32$ with $L=100^{{\prime}}$ in Example \ref{['numex:nclar3']}.
  • Figure 3: A realisation of a sample path of the FitzHugh-Nagumo model as specified in Equation \ref{['eq:fitz']}, with parameter values as in \ref{['eq:fitz-parvalues']}.
  • Figure 4: Realisations of $100$ forward sampled paths for the FitzHugh-Nagumo model as specified in Equation \ref{['eq:fitz']}, with parameter values as in \ref{['eq:fitz-parvalues']}.
  • Figure 5: Sampled guided diffusion bridges when conditioning on $v=-1$ (typical case).
  • ...and 6 more figures

Theorems & Definitions (48)

  • Proposition 2.3
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Proposition 2.8
  • Remark 2.9
  • Corollary 2.10: Uniformly elliptic case
  • Remark 2.11
  • Example 2.12
  • ...and 38 more