Doob-Meyer Decomposition

The Doob-Meyer theorem uniquely decomposes any square-integrable special semimartingale into a predictable finite-variation drift and a zero-mean local martingale. The Doob-Meyer decomposition does not apply literally to fractional Brownian motion with Hurst exponent H ≠ 0.5 because fBm is not a semimartingale in that regime. The Doob-Meyer decomposition provides a principled inductive bias for neural architecture design rather than serving only as an analytical description of the data-generating process. Standard neural operators trained with L2 loss implicitly recover only the drift component of the Doob-Meyer decomposition, leaving the martingale residual unmodeled. The Doob-Meyer split as an architectural prior suggests a general principle that classical stochastic process theorems can be encoded as neural network inductive biases.