Fast Dual Subgradient Optimization of Stochastic Kernels with Application to Forward-Backward Stochastic Differential Equations

A generalization of the Wasserstein metric, the integrated transportation distance, establishes a novel distance measure between probability kernels of Markov systems. This metric is the basis for an efficient approximation technique, allowing for substituting the original system’s kernel with a kernel having discrete support of limited cardinality. To facilitate implementation, we introduce a specialized dual algorithm capable of swiftly and efficiently constructing these approximate kernels, eliminating the need for computationally expensive matrix operations. Finally, we demonstrate our method’s effectiveness by approximating forward backward stochastic differential equations, highlighting its utility in practical financial scenarios. This advancement opens up new possibilities for streamlined analysis and manipulation of stochastic systems represented by kernels.