Self-normalized scaled quadratic variation

Abstract

The concept of a scaled quadratic variation was originally introduced by E. Gladyshev in 1961 for processes with Gaussian increments. Using certain deterministic scaling, arrived at from the covariance of the process, Gladyshev showed that the sum of scaled square increments along the dyadic partition sequence converges almost surely to a finite limit. In this paper, we propose a pathwise counterpart in which the deterministic normalization is replaced by a self-normalizing factor built from the $p$-th variation of the path along a given sequence of partitions. The resulting quantity requires no probabilistic assumption and no knowledge of a covariance structure, and its scale is both path-dependent and sensitive to the partition sequence. Under a mild regularity condition on the limiting $p$-th variation, we show that the self-normalized and the classical deterministic normalizations are comparable, and for fractional Brownian motion the two agree up to a multiplicative constant. We establish a switching behaviour in the index, and prove that for $p \ge 2$ the self-normalized scaled quadratic variation obeys a smooth-transformation formula under $C^2$ maps; at $p=2$ this recovers the known transformation rule for quadratic variation. Since only squared increments are scaled, the construction polarizes, yielding a matrix-valued scaled quadratic variation for every $p \geq 1$ for $\mathbb R^d$ valued paths. We conclude with examples beyond the Gaussian setting.