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For a family of stochastic differential equations, we investigate theasymptotic behaviors of its corresponding Picard's iteration, establishingconvergence results in terms of relative entropy. Our convergence resultscomplement the conventional ones in the $L^2$ and almost sure sense, revealingsome previously unexplored aspects of the stochastic differential equationsunder consideration. For example, in combination with Pinsker's inequality, oneof our results readily yields the convergence under Picard's iteration in thetotal variation sense, which does not seem to directly follow from any otherknown results. Moreover, our results promise possible further applications ofSDEs in related disciplines. As an example of such applications, we establishthe convergence of the corresponding mutual information sequence under Picard'siteration for a continuous-time Gaussian channel with feedback, which may paveway for effective computation of the mutual information of such a channel, along open problem in information theory.

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