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

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.