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Skew-Normal Diffusions

Max-Olivier Hongler, Daniele Rinaldo

TL;DR

This work constructs a class of diffusion processes driven by Gaussian white noise whose transition densities are skew-Normal, achieved via an $h$-transform and a dynamic censoring interpretation. It presents two complementary constructions—the $h$-transformed framework and a bi-dimensional Gaussian marginalization—that produce SKN processes with explicit skewed TP densities and finite-horizon support. The paper establishes key properties, including moment structure, linear-invariance under transformations, and a Brownian representation as a superposition of SKN components, enabling exact finite-dimensional nonlinear filtering that extends the Kalman-Bucy paradigm. These results provide a tractable, non-Gaussian diffusion model with potential applications in stochastic modeling and nonlinear filtering where skewness is essential. The formal tools and propositions offer a coherent link between measure-change techniques, dynamic censoring, and practical filtering methods for SKN-driven systems.

Abstract

We construct a class of stochastic differential equations driven by White Gaussian noise sources whose solutions can be drawn from skewed Gaussian probability laws, here referred as skew-Normal diffusion (SKN) processes. The non-Gaussian character results from implementing a nonlinear and time-inhomogneous drift constructed via ad-hoc changes of probability measure (i.e. Doob's $h$-transform). The SKN processes can be alternatively constructed as dynamic censoring models. While explicitly non-Gaussian, the SKN processes share several properties of Gaussian processes, in particular the invariance under linear transformations. This result allows us to discuss analytically the characteristics of this class of stochastic dynamics. As an illustration, we show how linear noisy monitoring of SKN processes yields a solvable finite dimensional and non-linear stochastic filtering which naturally extends the Kalman-Bucy Gaussian case.

Skew-Normal Diffusions

TL;DR

This work constructs a class of diffusion processes driven by Gaussian white noise whose transition densities are skew-Normal, achieved via an -transform and a dynamic censoring interpretation. It presents two complementary constructions—the -transformed framework and a bi-dimensional Gaussian marginalization—that produce SKN processes with explicit skewed TP densities and finite-horizon support. The paper establishes key properties, including moment structure, linear-invariance under transformations, and a Brownian representation as a superposition of SKN components, enabling exact finite-dimensional nonlinear filtering that extends the Kalman-Bucy paradigm. These results provide a tractable, non-Gaussian diffusion model with potential applications in stochastic modeling and nonlinear filtering where skewness is essential. The formal tools and propositions offer a coherent link between measure-change techniques, dynamic censoring, and practical filtering methods for SKN-driven systems.

Abstract

We construct a class of stochastic differential equations driven by White Gaussian noise sources whose solutions can be drawn from skewed Gaussian probability laws, here referred as skew-Normal diffusion (SKN) processes. The non-Gaussian character results from implementing a nonlinear and time-inhomogneous drift constructed via ad-hoc changes of probability measure (i.e. Doob's -transform). The SKN processes can be alternatively constructed as dynamic censoring models. While explicitly non-Gaussian, the SKN processes share several properties of Gaussian processes, in particular the invariance under linear transformations. This result allows us to discuss analytically the characteristics of this class of stochastic dynamics. As an illustration, we show how linear noisy monitoring of SKN processes yields a solvable finite dimensional and non-linear stochastic filtering which naturally extends the Kalman-Bucy Gaussian case.
Paper Structure (13 sections, 68 equations, 1 figure)

This paper contains 13 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. H-transforms and skewed diffusive Markov processes
  3. A particular class of measure changes
  4. Skew-Normal Diffusions
  5. Alternative constructions of skew-Normal diffusions
  6. The SKN as a truncated marginal of bi-dimensional Gaussian process
  7. SKN and dynamic censoring
  8. Properties of the SKN diffusion process
  9. Moments of the SKN diffusion process
  10. Linear transformations of the ESKN processes
  11. The BM as the superposition of SKN diffusions
  12. Illustration - finite dimensional nonlinear stochastic filtering
  13. SKN and finite-dimensional nonlinear filters

Figures (1)

  • Figure 1: Transient of the right-skewed TPD $p_{\pm}(x,t|0, 0)$ as given by Eq.(\ref{['TPDSKEW']}) for the time horizon $T=10$ and increasing time $t$ from $0.5$ to $T$ (thick red curve). By definition of the the Heaviside step function $\mathbbm{1}(x)$, we have $\lim_{t \rightarrow T} \Phi\left( \frac{\sqrt{2}}{\sqrt{T-t}} \right): = \mathbbm{1}(x)$ thus leading to the half-Normal law (red curve).