Skew-Normal Diffusions

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.