Occupation Times at a Moving Interface and Sharp Yosida Rates for MacKean-Vlasov BSDE with Jumps and Fuzzy Selections

Source abstract

We study a forward-backward McKean-Vlasov system with jumps whose backward equation contains a source obtained by evaluating the minimal-norm selection of a maximal monotone operator along the forward state. This selection changes branch across an interface that moves with the state law, the control, and an imprecisely known parameter described by fuzzy cuts. Assuming normal non-degeneracy and finite-statistics dependence of the interface on the law, we prove that the first and second moments of the time spent within a given distance of the interface are bounded, respectively, by a constant times that distance and by a constant times its square. The second-moment estimate makes it possible to integrate the source error before squaring it, which yields a Yosida error of first order in the regularization parameter in the natural solution norm; a first-moment argument gives only order one half. An explicit example shows that the first-order rate is sharp. For compact classes of admissible selections, the corresponding solutions form the cuts of a fuzzy solution set. These cuts are compact, stable with respect to the fuzzy level and the data, and approximated at the same first-order rate in Hausdorff distance. All estimates are uniform over a non-dominated family of probability measures representing volatility uncertainty. The quantitative results are obtained for separable quadratic potentials with finitely many affine branches.