Nonlinear Feynman–Kac formula and discrete-functional-type BSDEs with continuous coefficients

In this paper, we study a class of multi-dimensional backward stochastic differential equations (BSDEs, for short) in which the terminal values and the generators are allowed to be “discrete-functionals” of a forward diffusion. We first establish some new types of Feynman–Kac formulas related to such BSDEs under various regularity conditions, and then we prove that under only bounded continuous assumptions on the generators, the adapted solution to such BSDEs does exist. Our result on the existence of the solutions to higher-dimensional BSDEs is new, and our representation theorem is the first step towards the long-standing “functional-type” Feynman–Kac formula.