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 to as skew-Normal diffusion (SKN) processes. The non-Gaussian nature of such processes results from introducing a nonlinear and time-inhomogeneous drift constructed via ad-hoc changes of probability measure (Doob’s h-transform). SKN processes fit naturally within the statistical mechanics of trajectories as they are the driven processes associated with conditioning a Brownian motion on a terminal restriction to a subset of its domain. A SKN process can be alternatively constructed as a truncated marginal of a bi-dimensional diffusion, and can be interpreted as a dynamic censoring model. While explicitly non-Gaussian, 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 novel class of stochastic dynamics. As an illustration, we show how linear noisy monitoring of SKN processes yields a fully solvable, finite-dimensional and non-linear stochastic filter which naturally extends the Kalman-Bucy Gaussian case.