Skew-Product Decomposition of Planar Brownian Motion and Complementability

International audienceLet $Z$ be a complex Brownian motion starting at 0 and $W$ the complex Brownian motion defined by $W_ t = \int_0^\cdot \frac{Z_s}{|Z_s|} dZ_s$. The natural filtration $\mathcal{F}_W$ of $W$ is the filtration generated by $Z$ up to an arbitrary rotation. We show that given any two different matrices $Q_1$ and $Q_2$ in $O_2(\mathbb{R})$, there exists an $\mathcal{F}_Z$-previsible process $H$ taking values in $\{Q_1,Q_2\}$ such that the Brownian motion $\int_0^\cdot H \cdot dW$ generates the whole filtration $\mathcal{F}_Z$. As a consequence, for all $a$ and $b$ in $\mathbb{R}$ such that $a^2 + b^2 = 1$, the Brownian motion $a \mathrm{Re}(W) + b \mathrm{Im}(W)$ is complementable in $\mathcal{F}_Z$

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page

Functional inequalities on manifolds with non-convex boundary

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary

Langevin diffusions on the torus: estimation and applications

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein–Uhlenbeck process. Their likelihood function is a product of transition densities with no analytical expression, but that can be calculated by solving the Fokker–Planck equation numerically through adequate schemes. We propose three approximate likelihoods that are computationally tractable: (i) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and Shoji–Ozaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate transition densities to the exact ones, and investigates the empirical performance of the approximate likelihoods. Finally, two diffusions are used to model the evolution of the backbone angles of the protein G (PDB identifier 1GB1) during a molecular dynamics simulation. The software package sdetorus implements the estimation methods and applications presented in the paper