On Copula-Itô processes

Abstract

We study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

Keywords

• copulas
• copula processes
• stochastic differential equations
• parabolic partial differential equations
• semigroups of transformations
• 62h05
• 60j60
• 60h10
• 46e35
• 47h20
• 58d07