Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.1796

Representation Theorems for Quadratic ${\cal F}$-Consistent Nonlinear Expectations

Ying Hu·Jin Ma·Shige Peng·Song Yao
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we extend the notion of ``filtration-consistent nonlinear expectation" (or "${\cal F}$-consistent nonlinear expectation") to the case when it is allowed to be dominated by a $g$-expectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including the Doob-Meyer type decomposition theorem and the optional sampling theorem. More importantly, we show that any quadratic ${\cal F}$-consistent nonlinear expectation with a certain domination property must be a quadratic $g$-expectation. The main contribution of this paper is the finding of the domination condition to replace the one used in all the previous works, which is no longer valid in the quadratic case. We also show that the representation generator must be deterministic, continuous, and actually must be of the simple form.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ying Hu, Jin Ma, Shige Peng, Song Yao. 2007-04-13. Representation Theorems for Quadratic ${\cal F}$-Consistent Nonlinear Expectations. https://arxiv.org/abs/0704.1796

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random $3$-regular graphs are not globally synchronizing

We study the homogeneous Kuramoto model on random 3-regular graphs. A connected graph $G$ is globally synchronizing if, for almost every initial condition, the gradient flow converges to a fully synchronized state, in which all phases coincide. It has been conjectured by Bandeira [Oberwolfach Rep. 21 (2024) 1163-1226] that a random 3-regular graph $G_{2n,3}$ is globally synchronizing with high probability. We disprove this conjecture in a strong way: $G_{2n,3}$ is not globally synchronizing with probability tending to 1.

math.PR↗
arXiv2026

Poisson contingency tables, Gibbs permutations, and entropic optimal transport

We study random contingency tables obtained by conditioning a matrix of independent Poisson entries on its row and column sums. We identify such a table with the block counts of a random permutation drawn from a Gibbs measure of Mallows type, an exact coupling whose normalizing constants recover Barvinok's permanent identity. We use this correspondence to study the table at three scales. For finite dimensions, a transference principle compares the table with a matrix of independent Poisson entries whose means form the static Schrödinger bridge, the solution of a discrete entropic optimal transport problem. As the dimensions grow and the margins and prior means converge, the normalized histogram of the table converges to a limiting Schrödinger bridge at a quantitative rate. The fluctuations of the table around the bridge are characterized in two complementary ways. Linear statistics satisfy a central limit theorem for bounded test functions, with limiting variance equal to the squared $L^2$-distance, under the bridge, of the test function to the subspace of additively separable functions; as a process they converge to white noise with intensity given by the bridge, conditioned to have vanishing marginals. Spectrally, the limiting spectral distribution of the centered table is governed by a Dyson equation whose variance profile is the density of the bridge. The key ingredient is an asymptotic formula for the partition function of the Gibbs measure in terms of a Fredholm determinant given by the canonical correlations of the bridge. For Mallows-type random permutations, our results prove Pal's conjecture on the limiting partition function and Mukherjee's conjecture on the central limit theorem for linear statistics, both in greater generality.

math.PR↗
arXiv2026

Large deviation principle at speed $n^d$ for the random metric in first-passage percolation

We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded weights. We show a large deviation estimate for the probability that the rescaled passage time restricted to a box is greater than a target pseudometric. Moreover, the associated rate function can be written as the integral over the box of an elementary cost. Under the additional assumption that weights are bounded away from zero, the random pseudometric follows a large deviation principle (LDP) at speed $n^d$. Consequences include LDPs at speed $n^d$ for the point-to-point passage time, the crossing passage time and the random ball of radius $n$. Our strategy consists in proving first the existence of the rate function for any seminorm $g$ with a multidimensional subadditivity argument, then using this result as an elementary building block to estimate the rate function in full generality.

math.PR↗