Radon-Nikodym Research Articles

Separable representations, KMS states, and wavelets for higher-rank graphs

Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C⁎(Λ) on certain separable Hilbert spaces of the form L2(X,μ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C⁎(Λ) on L2(Λ∞,M), where M is the Perron–Frobenius probability measure on the infinite path space Λ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C⁎(Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of C⁎(Λ) on L2(X,μ), where X is a fractal subspace of [0,1] by embedding Λ∞ into [0,1] as a fractal subset X of [0,1]. Moreover, when the Radon–Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C⁎(Λ). Finally, we construct a wavelet system for L2(Λ∞,M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

Minimum divergence estimators for the Radon–Nikodym derivatives of the semi-Markov kernel

In this paper we face the problem of estimating the elements that define a homogeneous semi-Markov process. Given an observed sample path of the process in a finite interval , non-parametric estimators of the entries of the semi-Markov kernel and its derivatives can be constructed. These estimators exhibit good properties as consistency and asymptotic normality. On the other hand, in our approach we consider that we have some prior information about the underlying process. This information concerns the mean sojourn times in the different states of the process and has to be taken into account in the estimation procedure. So we start from the smoothed estimators and construct minimum divergence estimators (maximum entropy) under some constraints for the associated sojourn moments.

On the f-divergence for non-additive measures

The f-divergence evaluates the dissimilarity between two probability distributions defined in terms of the Radon–Nikodym derivative of these two probabilities. The f-divergence generalizes the Hellinger distance and the Kullback–Leibler divergence among other divergence functions. In this paper we define an analogous function for non-additive measures. We discuss them for distorted Lebesgue measures and give examples. Examples focus on the Hellinger distance.

Inverting Ray-Knight identity

We provide a short proof of the Ray-Knight second generalized Theorem, using a martingale which can be seen (on the positive quadrant) as the Radon–Nikodym derivative of the reversed vertex-reinforced jump process measure with respect to the Markov jump process with the same conductances. Next we show that a variant of this process provides an inversion of that Ray-Knight identity. We give a similar result for the Ray-Knight first generalized Theorem.

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron–Martin subgroup, and that the Radon–Nikodym derivative is Malliavin smooth.

Unbounded subnormal composition operators in L2-spaces

A novel criterion for subnormality of unbounded composition operators in L2-spaces, written in terms of measurable families of probability measures satisfying the so-called consistency condition, is established. It becomes a new characterization of subnormality in the case of bounded composition operators. Pseudo-moments of a measurable family of probability measures that satisfies the consistency condition are proved to be given by the Radon–Nikodym derivatives which appear in Lambert's characterization of bounded composition operators. A criterion for subnormality of composition operators induced by matrices is provided. The question of subnormality of composition operators over discrete measure spaces is studied. Two new classes of subnormal composition operators over discrete measure spaces are introduced. A recent criterion for subnormality of weighted shifts on directed trees by the present authors is essentially improved in the case of rootless directed trees and nonzero weights by dropping the assumption of density of C∞-vectors in the underlying ℓ2-space.

Equivalence of Palm measures for determinantal point processes associated with Hilbert spaces of holomorphic functions

We obtain explicit formulae, in the form of regularized multiplicative functionals related to certain Blaschke products, of the Radon–Nikodym derivatives between reduced Palm measures of all orders for determinantal point processes associated with a large class of weighted Bergman spaces on the disk. Our method also applies to determinantal point processes associated with weighted Fock spaces.

On Banach–Mazur limits

A positive functional x∗ on the space ℓ∞ of all bounded sequences is called a Banach–Mazur limit if ‖x∗‖=1 and x∗x=x∗Tx for all x=(x1,x2,…)∈ℓ∞, where T is the forward shift operator on ℓ∞, i.e., Tx=(0,x1,x2,…). The set of all Banach–Mazur limits is denoted by BM and a collection of extreme points of BM is denoted by extBM. Let ac0={x∈ℓ∞:x∗x=0for allx∗∈BM}. The following sequence spaces D(ac0)={x∈ℓ∞:x⋅ac0⊆ac0}andI(ac0)=ac0+−ac0+ are studied. In particular, if z∈ℓ∞ then z∈D(ac0) iff z−Tz∈I(ac0); moreover, z∈D(ac0) iff x∗{n:|zn−x∗z|≥ϵ}=0 for all ϵ>0 and x∗∈extBM. Order properties of Banach–Mazur limits are considered. Some properties of extBM are derived. We used the representation of functionals x∗∈BM as Borel measures on βN∖N. The cardinalities of some subset of BM are given. We also consider some questions of the probability theory for finite additive measures. E.g., for every x∗∈BM there exists an element x∈ℓ∞ such that the distribution function Fx∗,x(t)=x∗{n:xn≤t} is continuous on R. Two definitions of a variance are suggested. It is shown that Radon–Nikodym theorem is not valid for finite additive measures: the relations 0≤x∗≤y∗∈ℓ∞∗ do not imply the existence of w∈ℓ∞ satisfying x∗x=y∗(wx) for all x∈ℓ∞.

Teaching Measure Transformation in Option Pricing with a Loaded Die

We argue that students can be introduced to the counter-intuitive concept of probability measure transformation by applying it to gambling games with a die. The effect of loading a die on the odds of a simple game is readily understood by students and transformed probabilities can easily be calculated. More complex transformations can be obtained with the solver of a spreadsheet program. They can be expressed in measure transformation functions that give an intuitive understanding of the Radon-Nikodym derivative used in risk neutral valuation. The connection with option pricing is made by placing a die game in a simple but general pricing framework. The die loading analogy may be of use to instructors teaching risk neutral valuation or probability measure transformation.

A Double Correlated Three Factor Model for a Crude Oil Market

In this paper we analyze spot prices and futures quotation data to get inference under both the historical and the risk neutral measure in a commodity crude oil market (data are referred to WTI index which tracks the crude oil barrel price on NYMEX market). While big part of research and techniques in finance deals with risk neutral modeling or with modeling under the historical measure, the purpose of this work is to study the estimation problem under both measures at the same time, through a suitable parametric choice of the Radon-Nikodym derivative. To conduct this estimation we resort to a recent technique in Bayesian inference: the Particle Markov Chain Monte Carlo proposed by Andrieu, Doucet and Holenstein (2010), in which particle filters algorithms are used to estimate the marginal likelihood needed to perform a Markov Chain Monte Carlo inference. We adopt a three factors model to describe the dynamics of the spot price, including a stochastic dynamics for both the volatilty and the convenience yield. For the present model the futures prices and future options prices are available in closed form. Two versions of the original model, with and without jumps in spot prices, are taken into account and results are compared. A comparison is also provided between these models and that originally proposed by Liu and Tang (2011). The inference analysis shows that volatility and jumps improve significantly the ability of the model in capturing both the risk-neutral and the historical dynamics features.

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).

Estimation of Value-at-Risk with Minimal Specification Error

In this paper we provide a new statistical approach to estimate VaR with minimal specification error. In our approach we use current market information obtained from traded options to infer risk-neutral moments of the underlying asset returns over different horizons. For each horizon, the moments are fitted to a general 4-moment Variance Gamma risk-neutral probability distribution of future market prices. We also compute a Radon-Nikodym derivative and its probability distribution, and use this to transform the risk-neutral distribution of the underlying to its empirical distribution. The empirical distribution is then used to find the VaR. The novelty of our approach is that it avoids unnecessary mis-specification bias as we do not need to assume a particular empirical distribution of the underlying asset returns. As option prices trade frequently, the risk-neutral distribution can be updated frequently for transformation into the empirical distribution. This provides for timely updating of the VaR measures. The fitted risk-neutral Variance Gamma distribution is popular in finance due to its ability to model heavy tails and skewness. Our key contributions are in providing an alternative new approach to modeling VaR, and also in showing that underestimation of risk is largely not due to VaR itself but perhaps due to mis-specification errors which we minimize in our approach. We show that our method of measuring VaR clearly captures large tail risk in the empirical examples on S&P 500 index.

Continuous dependence on the derivative of generalized heat equations

Here, we consider a generalized heat equation $\partial_t \rho=\frac{d}{dx}\frac{d}{dW}\rho$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such an equation has appeared in different contexts, one of which being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists of proving, for a suitable topology, a continuous dependence of the solution $\rho$ as a function of $W$.

Regularity of densities in relaxed and penalized average distance problem

The average distance problem finds application in data parameterization, which involves``representing'' the data using lower dimensional objects. Froma computational point of view it is often convenient to restrictthe unknown to the family of parameterized curves.The original formulation of the average distance problem exhibits several undesirable properties. In this paper we propose an alternative variant: we minimize the functional \begin{equation*}\int_{{\mathbb{R}}^d\times \Gamma_\gamma} |x-y|^p {\,{d}}\Pi(x,y)+\lambda L_\gamma +\varepsilon\alpha(\nu)+\varepsilon' \eta(\gamma)+\varepsilon''\|\gamma'\|_{TV},\end{equation*}where $\gamma$ varies among the family of parametrized curves, $\nu$ among probability measures on $\gamma$,and $\Pi$ among transport plans between $\mu$ and $\nu$. Here $\lambda,\varepsilon,\varepsilon',\varepsilon''$ are given parameters,$\alpha$ is a penalization term on $\mu$, $\Gamma_\gamma$ (resp. $L_\gamma$) denotes the graph (resp. length)of $\gamma$, and $\|\cdot\|_{TV}$ denotes the total variation semi-norm.We will use techniques from optimal transport theory and calculus of variations. The main aim is to prove essential boundedness, and Lipschitz continuity for Radon-Nikodym derivative of $\nu$, when $(\gamma,\nu,\Pi)$is a minimizer.

Radon–Nikodym Theorems for Nonnegative Forms, Measures and Representable Functionals

The aim of this paper is to establish two Radon–Nikodym-type theorems for nonnegative Hermitian forms defined on a real or complex vector space and to apply these results to provide some known Radon–Nikodym-type theorems of the theory of representable positive functionals, \(\sigma \)-additive and finitely additive measures.

A Pre-Trade Algorithmic Trading Model Under Given Volume Measures and Generic Price Dynamics

We make several improvements to the mean–variance framework for optimal pre-trade algorithmic execution, by working with volume measures and generic price dynamics. Volume measures are the continuum analogies for discrete volume profiles commonly implemented in the execution industry. Execution then becomes an absolutely continuous measure over such a measure space, and its Radon–Nikodym derivative is commonly known as the participation of volume (PoV) function. The four impact cost components are all consistently built upon the PoV function. Some novel efforts are made for these linear impact models by having market signals more properly expressed. For the opportunistic cost, we go beyond the conventional Brownian-type motions. By working directly with the auto-covariances of the price dynamics, we remove the Markovian restriction associated with Brownians and thus allow potential memory effects. In combination, the final execution model becomes a constrained quadratic programming problem in infinite-dimensional Hilbert spaces. Important linear constraints such as participation capping are all permissible. Uniqueness and existence of optimal solutions are established via the theory of positive compact operators in Hilbert spaces. Several typical numerical examples explain both the behavior and versatility of the model.

Bi-invariant functions on the group of transformations leaving a measure quasi-invariant

Let be the group of transformations of a Lebesgue space leaving the measure quasi-invariant. Let be a subgroup of it consisting of transformations preserving the measure. We describe canonical forms of double cosets of by the subgroup and show that all continuous -bi-invariant functions on are functionals of the distribution of a Radon-Nikodym derivative. Bibliography: 14 titles.

Semiclassical measures for the Schrödinger equation on the torus

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.

Bayes Hilbert Spaces

SummaryA Bayes linear space is a linear space of equivalence classes of proportional σ‐finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes' rule and substraction by Radon–Nikodym derivatives. The present contribution shows the subspace of square‐log‐integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred‐log‐ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square‐integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support.

The idempotent Radon–Nikodym theorem has a converse statement

Idempotent integration is an analog of the Lebesgue integration where σ-additive measures are replaced by σ-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon–Nikodym derivatives, which turns out to be crucial in all these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon–Nikodym theorem, i.e. we characterize the σ-maxitive measures that have the Radon–Nikodym property.