Radon-Nikodym Research Articles

Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.

On the Calibration of the Kennedy Model

The Kennedy model offers a robust framework for modeling forward rates, leveraging Gaussian random fields to accommodate emerging phenomena such as negative rates. In our study, we employ maximum likelihood estimations to determine the parameters of the Kennedy field, utilizing Radon–Nikodym derivatives for enhanced accuracy. We introduce an efficient simulation method for the Kennedy field and develop a Black–Scholes-like analytical pricing formula for diverse financial assets. Additionally, we present a novel parameter estimation algorithm grounded in numerical extreme value optimization, enabling the recalibration of parameters based on observed financial product prices. To validate the efficacy of our approach, we assess its performance using real-world par swap rates in the latter part of this article.

Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.

Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula

Let ρ be compactly supported on D⊂R2. Endow R2 with the metric eρ|dx|2. As δ→0 the set of Brownian loops centered in D with length at least δ has measureVol(D)2πδ+148π(ρ,ρ)∇+o(1) where (ρ,ρ)∇=∫D|∇ρ(x)|2dx. When ρ is smooth, this follows from the classical Polyakov-Alvarez formula. We show that the above also holds if ρ is not smooth, e.g. if ρ is only Lipschitz. Variants of this statement apply to more general non-smooth manifolds on which one considers all loops (not only those centered in a domain D).We also show that the o(1) error is uniform for any family of ρ satisfying certain conditions. This implies that if we weight a measure ν on this family by the (δ-truncated) Brownian loop soup partition function, and take the vague δ→0 limit, we obtain a measure whose Radon-Nikodym derivative with respect to ν is exp⁡(148π(ρ,ρ)∇). When the measure is a certain regularized Liouville quantum gravity measure, a companion work [5] shows that this weighting has the effect of changing the so-called central charge of the surface.

Extension theorems of additive interval multifunctions with applications

In this paper, we present extension theorems of additive interval multifunctions to multimeasures in locally convex topological vector spaces. Applying these extension results, we present Radon-Nikodym theorems for additive interval multifunctions.

Gauge Field Marginal of an Abelian Higgs Model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon–Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler (Gauge theories as a problem of constructive quantum field theory and statistical mechanics, volume 159 of lecture notes in physics, Springer, Berlin, p v+192. https://doi.org/10.1007/3-540-11559-5, 1982) but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the Hölder–Besov-type norms introduced in Chevyrev (Commun Math Phys 372(3):1027–1058. https://doi.org/10.1007/s00220-019-03567-5, 2019) are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.

New neighborhoods in robustness and least favorable pairs for special capacities

We propose a new neighborhood of distributions for robust inference, which is generated by a certain special capacity with three constants. It covers not only the familiar neighborhoods defined in terms of ε-contamination, total variation and their combination, but also their intersections. It is shown that the neighborhood is also obtained from contaminating some neighborhood which consists of absolutely continuous distributions. We derive the explicit forms of Radon-Nikodym derivatives and the least favorable pairs for such special capacities and neighborhoods on the real line, which generalize the earlier results of Huber and Rieder for minimax testing, and reinforce the general results of Bednarski.

A Radon-Nikodym theorem for monotone measures

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair (μ,ν) of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function f. If ν is null-continuous and weakly null-additive, then f is uniquely determined almost everywhere by ν and thus is called the Radon-Nikodym derivative of μ w.r.t. ν. For σ-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

On existence of Berk-Nash equilibria in misspecified Markov decision processes with infinite spaces

Model misspecification is a critical issue in many areas of economics. In the context of misspecified Markov Decision Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium and established its existence with finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo) involve continuous state or action spaces, and are thus not covered by the Esponda-Pouzo existence theorem. We extend the existence of Berk-Nash equilibrium to compact action spaces and sigma-compact state spaces, with possibly unbounded utility functions. A complication arises because Berk-Nash equilibrium depends critically on Radon-Nikodym derivatives, which are bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis, and draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

A posterior contraction for Bayesian inverse problems in Banach spaces

This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon–Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error. Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.

Optimal investment–consumption–insurance strategy with inflation risk and stochastic income in an Itô–Lévy setting

This paper’s focus is on finding the optimal strategies for a trader who invests in stock, a money market account and an inflation-linked index bond. The stock follows a jump diffusion process and the bond is linked to inflation making the two risky. The optimal strategies are determined on two generations of the life of an investor, that is before the investor dies and after the investor dies. We applied the concept of change of probability measures considering Girsanov’s and the Radon–Nikodym theorems. We found the generator of the Backward Stochastic differential equations defined and employed the Hamilton–Jacobi–Bellman (HJB) dynamic programming in finding the stochastic optimal controls of interest.

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon–Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

Nagy type inequalities in metric measure spaces and some applications

We obtain a sharp Nagy type inequality in a metric space $(X,\rho)$ with measure $\mu$ that estimates the uniform norm of a function using its $\|\cdot\|_{H^\omega}$-norm determined by a modulus of continuity $\omega$, and a seminorm that is defined on a space of locally integrable functions. We consider charges $\nu$ that are defined on the set of $\mu$-measurable subsets of $X$ and are absolutely continuous with respect to $\mu$. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a $\|\cdot\|_{H^\omega}$-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case $X={\mathbb R}_+^m\times {\mathbb R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\|\cdot\|_{H^\omega}$-norm of its mixed derivative.

Statistical evaluation of a long‐memory process using the generalized entropic value‐at‐risk

AbstractThe modeling and identification of time series data with a long memory are important in various fields. The streamflow discharge is one such example that can be reasonably described as an aggregated stochastic process of randomized affine processes where the probability measure, we call it reversion measure, for the randomization is not directly observable. Accurate identification of the reversion measure is critical because of its omnipresence in the aggregated stochastic process. However, the modeling accuracy is commonly limited by the available real‐world data. We resolve this issue by proposing the novel upper and lower bounds of a statistic of interest subject to ambiguity of the reversion measure. Here, we use the Tsallis value‐at‐risk (TsVaR) as a convex risk functional to generalize the widely used entropic value‐at‐risk (EVaR) as a sharp statistical indicator. We demonstrate that the EVaR cannot be used for evaluating key statistics, such as mean and variance, of the streamflow discharge due to the blowup of some exponential integrand. We theoretically show that the TsVaR can avoid this issue because it requires only the existence of some polynomial moment, not exponential moment. As a demonstration, we apply the semi‐implicit gradient descent method to calculate the TsVaR and corresponding Radon–Nikodym derivative for time series data of actual streamflow discharges in mountainous river environments.

Asymptotics for Christoffel functions associated to continuum Schrödinger operators

We prove asymptotics of the Christoffel function, λL(ξ), of a continuum Schrödinger operator for points in the interior of the essential spectrum under some mild conditions on the spectral measure. It is shown that LλL(ξ) has a limit and that this limit is given by the Radon–Nikodym derivative of the spectral measure with respect to the Martin measure. Combining this with a recently developed local criterion for universality limits at scale λL(ξ), we compute universality limits for continuum Schrödinger operators at scale L and obtain clock spacing of the eigenvalues of the finite range truncations.

The Donsker delta function and local time for McKean–Vlasov processes and applications

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean–Vlasov (mean-field) stochastic differential equation.If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon–Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process.For some particular McKean–Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

Stressing dynamic loss models

Stress testing, and in particular, reverse stress testing, is a prominent exercise in risk management practice. Reverse stress testing, in contrast to (forward) stress testing, aims to find an alternative but plausible model such that under that alternative model, specific adverse stresses (i.e. constraints) are satisfied. Here, we propose a reverse stress testing framework for dynamic models. Specifically, we consider a compound Poisson process over a finite time horizon and stresses composed of expected values of functions applied to the process at the terminal time. We then define the stressed model as the probability measure under which the process satisfies the constraints and which minimizes the Kullback-Leibler divergence to the reference compound Poisson model.We solve this optimization problem, prove existence and uniqueness of the stressed probability measure, and provide a characterization of the Radon-Nikodym derivative from the reference model to the stressed model. We find that under the stressed measure, the intensity and the severity distribution of the process depend on time and state, and hence the stressed model is not a compound Poisson process. We illustrate the dynamic stress testing by considering stresses on VaR and both VaR and CVaR jointly and provide illustrations of how the stochastic process is altered under these stresses. We generalize the framework to multivariate compound Poisson processes and stresses at times other than the terminal time. We illustrate the applicability of our framework by considering “what if” scenarios, where we answer the question: What is the severity of a stress on a portfolio component at an earlier time such that the aggregate portfolio exceeds a risk threshold at the terminal time? Furthermore, for general constraints, we propose an algorithm to simulate sample paths under the stressed measure, thus allowing to compare the effects of stresses on the dynamics of the process.

Orlicz risks for assessing stochastic streamflow environments: a static optimization approach

This study applies novel risk measures, called Orlicz risks, to the risk and uncertainty evaluation of the streamflow discharge as a primary driver of hydrological and hydraulic processes of interest in civil and environmental engineering. We consider the mixed moving average process governing the discharge whose statistics are explicitly represented as some product of a time-scale characterizing the flow attenuation and a jump moment governing the size and frequency of jumps. The classical Orlicz risks are extended so that not only the upper tail risk but also the lower one of the jump size and attenuation of the discharge can be evaluated within a single mathematical framework. Further, the risk and uncertainty can be individually quantified in a tractable manner by the proposed Orlicz risks. Computing the Orlicz risks reduces to solving a pair of novel static optimization problems that are solvable semi-analytically. The risk and uncertainty involved in the streamflow dynamics can be consistently evaluated by specifying few user-dependent parameters. The associated Radon–Nikodym derivatives as the worst-case model uncertainties are obtained as byproducts. Sufficient conditions for the well-posedness of the Orlicz risks are discussed and numerical algorithms for computing them are presented. We finally apply the proposed framework to a statistical analysis of the streamflow discharge time series data collected at mountainous river environments.

On Landau-Kolmogorov type inequalities for charges and their applications

In this article we prove sharp Landau-Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $\mathbb{R}^d$, $d\geqslant 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon-Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$ belongs to the Sobolev space $W^{1,\infty}$.