Robert Aebi

Improving Models of Income Dynamics using Cross-Section Information

SummaryBased on a relative entropy approach, this paper proposes a method to estimate or update transition matrices using just cross-sectional observations at two points in time. The method is then applied to explain the development of the US income distribution. Starting from three hypothesized transition matrices and a transition matrix estimated from the PSID data, we show how these matrices must be adjusted in the light of the cross-sectional information. Finally, we explore the consequences of these updated transition matrices for the future development of the US income distribution.

Schrödinger’s View of Natural Laws

This is an exploration with today’s tools of probability theory into the mystery of ‘Uber die Umkehrung der Naturgesetze’ (On time-reversal of natural laws) written by Erwin Schrodinger in 1931. He propagates the idea that diffusions given their marginal distributions at finite initial and final time are time-reversible. We are going to meet clouds of identical and independent particles considering them as realizations of diffusions. The particle dynamics is assumed to be known. Given the observation of such a cloud in terms of distribution densities at finite initial and final time, the intention is to find the ‘most probable’ distribution density of the cloud at intermediate times.

Interacting Diffusion Processes

The phenomenon ‘propagation of chaos’ is discussed in terms of the relative entropy in (5.26) which allows general microscopical systems and which provides results in the tradition of statistical mechanics. In Section 6.3 we show ‘propagation of chaos in entropy’ for particle clouds with prescribed initial and final distributions: The particles become asymptotically independent and perform identically according to a Csiszar projection as their number increases to infinity. These limiting distributions turn out to be Schrodinger processes, i.e., diffusion processes considered from Schrodinger (1931)’s time-symmetrical point of view. They are uniquely characterized by the large deviation principle deduced in Chapter 5. The associated rate function minimizes the relative entropy with respect to a renormalized Markovian reference process which has in general singular creation and killing as discussed in Sections 6.2 and 6.4. The n-product of the renormalized reference process conditioned by means of the empirical distribution on an approximation of A a,b in (5.2) possesses a Markovian modification. This system of interacting diffusion processes is proved to perform propagation of chaos in entropy with a Schrodinger process as limiting distribution.

Diffusions with Singular Drift

Schrodinger equations are shown in Section 2.1 to be equivalent to pairs of adjoint non-linear diffusion equations. Their weak solutions are treated in Section 2.2 as elements of the local space-time Sobolev space H loc 1,2 which is motivated from the L2-theory of Schrodinger equations.

Integral and Diffusion Equations

Let us investigate the correspondence of weak solutions of parabolic differential equations with singular creation and killing and solutions of Feynman-Kac integral equations with locally integrable potential. The correspondence is stated in Section 3.4 where the solutions are treated as locally integrable functions, continuity is neither required nor established. In Section 3.2 we meet an integral equation with a solution which can be given in terms of the Feynman-Kac formula. Stochastic calculus leads in Section 3.3 to a ‘killed’ integral equation which provides a refined uniqueness condition of solutions.