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Generator Identification for Linear SDEs with Additive and Multiplicative Noise

Yuanyuan Wang, Xi Geng, Wei Huang, Biwei Huang, Mingming Gong

TL;DR

This work derives identifiability conditions for the generator of linear SDEs from the observational distribution with a fixed initial state, enabling recovery of post-intervention distributions for causal analysis. For additive-noise SDEs, it provides a necessary-and-sufficient rank condition when the drift matrix $A$ has distinct eigenvalues, and shows a controllability-based corollary; for multiplicative-noise SDEs, it presents a sufficient (though not necessary) pair of conditions under an explicit-solution assumption. Both sets of conditions are shown to be generic and are given geometric interpretations via invariant subspaces and Jordan forms, with simulations validating the theoretical claims. The work connects generator identifiability to causal inference in dynamic systems and discusses practical limitations, such as parameter verification and computational cost, while pointing to avenues for efficient estimation and constrained modeling.

Abstract

In this paper, we present conditions for identifying the generator of a linear stochastic differential equation (SDE) from the distribution of its solution process with a given fixed initial state. These identifiability conditions are crucial in causal inference using linear SDEs as they enable the identification of the post-intervention distributions from its observational distribution. Specifically, we derive a sufficient and necessary condition for identifying the generator of linear SDEs with additive noise, as well as a sufficient condition for identifying the generator of linear SDEs with multiplicative noise. We show that the conditions derived for both types of SDEs are generic. Moreover, we offer geometric interpretations of the derived identifiability conditions to enhance their understanding. To validate our theoretical results, we perform a series of simulations, which support and substantiate the established findings.

Generator Identification for Linear SDEs with Additive and Multiplicative Noise

TL;DR

This work derives identifiability conditions for the generator of linear SDEs from the observational distribution with a fixed initial state, enabling recovery of post-intervention distributions for causal analysis. For additive-noise SDEs, it provides a necessary-and-sufficient rank condition when the drift matrix has distinct eigenvalues, and shows a controllability-based corollary; for multiplicative-noise SDEs, it presents a sufficient (though not necessary) pair of conditions under an explicit-solution assumption. Both sets of conditions are shown to be generic and are given geometric interpretations via invariant subspaces and Jordan forms, with simulations validating the theoretical claims. The work connects generator identifiability to causal inference in dynamic systems and discusses practical limitations, such as parameter verification and computational cost, while pointing to avenues for efficient estimation and constrained modeling.

Abstract

In this paper, we present conditions for identifying the generator of a linear stochastic differential equation (SDE) from the distribution of its solution process with a given fixed initial state. These identifiability conditions are crucial in causal inference using linear SDEs as they enable the identification of the post-intervention distributions from its observational distribution. Specifically, we derive a sufficient and necessary condition for identifying the generator of linear SDEs with additive noise, as well as a sufficient condition for identifying the generator of linear SDEs with multiplicative noise. We show that the conditions derived for both types of SDEs are generic. Moreover, we offer geometric interpretations of the derived identifiability conditions to enhance their understanding. To validate our theoretical results, we perform a series of simulations, which support and substantiate the established findings.
Paper Structure (28 sections, 13 theorems, 223 equations, 2 tables)

This paper contains 28 sections, 13 theorems, 223 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background knowledge
  3. Background knowledge of linear SDEs
  4. Causal interpretation of SDEs
  5. Main results
  6. Prerequisites
  7. Conditions for identifying generators of linear SDEs with additive noise
  8. Conditions for identifying generators of linear SDEs with multiplicative noise
  9. Simulations and examples
  10. Related work
  11. Conclusion and discussion
  12. Appendix for "Generator Identification for Linear SDEs with Additive and Multiplicative Noise"
  13. Detailed proofs
  14. Proof of Lemma \ref{['lemma:levy process']}
  15. Proof of Proposition \ref{['theorem:generator same']}
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let $X$ be a stochastic process defined by the SDE eq:diffusion process. The generator $\mathcal{L}$ of $X$ on $C_b^2(\mathbb{R}^d)$ is given by for $f \in C_b^2(\mathbb{R}^d)$ and $x \in \mathbb{R}^d$, where $c(x) = \sigma(x) \cdot \sigma(x)^{\top}$ is a $d\times d$ matrix, and $C_b^2(\mathbb{R}^d)$ denotes the space of continuous functions on $\mathbb{R}^d$ that have bounded derivatives up to o

Theorems & Definitions (29)

  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 3.1
  • Proposition 3.1
  • Definition 3.1: $(x_0, A, G)$-identifiability
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • ...and 19 more