Skew-Normal Diffusions

TL;DR

This paper introduces skew-Normal diffusion processes driven by Gaussian noise, constructed via nonlinear drifts and measure changes, which exhibit skewed Gaussian distributions while retaining some Gaussian properties, enabling analytical study and nonlinear filtering extensions.

Contribution

The paper develops a new class of stochastic differential equations called skew-Normal diffusion processes, expanding Gaussian models to include skewness while preserving linear invariance properties.

Findings

SKN processes can be explicitly constructed using measure changes.

They exhibit skewed Gaussian distributions with Gaussian-like invariance.

A nonlinear filtering extension of the Kalman-Bucy filter is derived.

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.