Relative entropy convergence under Picard's iteration for stochastic differential equations

TL;DR

This paper studies how solutions to stochastic differential equations converge under Picard's iteration using relative entropy, revealing new convergence insights and applications in information theory.

Contribution

It introduces convergence results in relative entropy for Picard's iteration of SDEs, including total variation convergence and applications to mutual information in Gaussian channels.

Findings

Convergence in relative entropy established for Picard's iteration.

Total variation convergence derived via Pinsker's inequality.

Mutual information sequence converges in a Gaussian channel with feedback.

Abstract

For a family of stochastic differential equations, we investigate the asymptotic behaviors of its corresponding Picard's iteration, establishing convergence results in terms of relative entropy. Our convergence results complement the conventional ones in the $L^2$ and almost sure sense, revealing some previously unexplored aspects of the stochastic differential equations under consideration. For example, in combination with Pinsker's inequality, one of our results readily yields the convergence under Picard's iteration in the total variation sense, which does not seem to directly follow from any other known results. Moreover, our results promise possible further applications of SDEs in related disciplines. As an example of such applications, we establish the convergence of the corresponding mutual information sequence under Picard's iteration for a continuous-time Gaussian channel with feedback, which may pave way for effective computation of the mutual information of such a channel, a long open problem in information theory.