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We introduce a distance between kernels based on the Wasserstein distancesbetween their values, study its properties, and prove that it is a metric on anappropriately defined space of kernels. We also relate it to various modes ofconvergence in the space of kernels. Then we consider the problem ofapproximating solutions to forward--backward systems, where the forward part isa Markov system described by a sequence of kernels, and the backward partcalculates the values of a risk measure by operators that may be nonlinear withrespect to the system's kernels. We propose to recursively approximate theforward system with the use of the integrated transportation distance betweenkernels and we estimate the error of the risk evaluation by the errors ofindividual kernel approximations. We illustrate the results on stoppingproblems and several well-known risk measures. Then we develop a particle-basednumerical procedure, in which the approximate kernels have finite support sets.Finally, we illustrate the efficacy of the approach on the financial problem ofpricing an American basket option.

An Integrated Transportation Distance Between Kernels and Approximate Dynamic Risk Evaluation in Markov Systems