Probabilistic deformation of contact geometry, diffusion processes and their quadratures

Classical contact geometry is an odd-dimensional analogue of symplectic geometry. We show that a natural probabilistic deformation of contact geometry, compatible with the very irregular trajectories of diffusion processes, allows one to construct the stochastic version of a number of basic geometrical tools, like, for example, Lionville measure. Moreover, it provides an unified framework to understand the origin of explicit relations (cf. "quadrature") between diffusion processes, useful in many fields. Various applications are given, including one in stochastic finance.