Third-moment cancellation and predictable representation in finite-state BSDE approximation

Abstract

We study finite-state approximations of Brownian-driven Markovian backward stochastic differential equations when the same one-step increment is used both for weak propagation and for martingale representation. For a centered isotropic state-independent increment with vanishing third moment, we prove first-order convergence of the Markov value field, its spatial gradient, and the predictable $Z$-field under suitable smoothness assumptions. The corresponding orthogonal martingale has maximal $L^2$ size $O(h^{1/2})$. We identify its quadratic representation defect and prove a joint functional limit theorem for the state process and the normalized orthogonal martingale, which in particular yields a mixed-normal terminal limit with path-dependent conditional variance. On the trace-free quadratic sector, the representation defect has the sharp universal operator lower bound $d/[2(d+2)]$; equality holds exactly for normalized laws that are weighted spherical 4-designs. In dimension $d\geq 2$, vanishing third moment is incompatible with exact scalar one-step predictable representation for any finite-state centered isotropic increment. Consequently, every such exact-representation law has a nonzero cubic direction, and a smooth Fourier terminal condition converts this direction into a sharp global $h^{1/2}$ weak error. For the centrally symmetric regular-simplex rule we also identify the leading quadratic GKW component, compute the exact first-order asymptotic of the accumulated orthogonal-martingale variance, and recover it from the general functional limit. Thus the same third-order tensor links weak accuracy and predictable representation, yielding an accuracy-representation tradeoff for finite-state BSDE approximations.