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 L2 size O(h1/2). 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≥ 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 h1/2 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.
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BibTeX
@misc{tanaka2026thirdmoment,
title = {{Third-moment cancellation and predictable representation in finite-state BSDE approximation}},
author = {Akihiro Tanaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20880},
eprint = {2610.20880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20880},
}
RIS
TY - UNPB
TI - Third-moment cancellation and predictable representation in finite-state BSDE approximation
AU - Akihiro Tanaka
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20880
UR - https://arxiv.org/abs/2610.20880
AB - 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≥ 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.
ER -