Reference Probability Measure Research Articles

Risk sharing under heterogeneous beliefs without convexity

We consider the problem of finding (Pareto-)optimal allocations of risk among finitely many agents. The associated individual risk measures are law-invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations, but remain agnostic about their convexity. A simple sufficient condition for the existence of Pareto optima is provided. The proof combines local comonotonic improvement with a Dieudonné-type argument, which also establishes a link of the optimal allocation problem to the realm of “collapse to the mean” results.

Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure

In this paper, the worst-case probability measure over the data is introduced as a tool for characterizing the generalization capabilities of machine learning algorithms. More specifically, the worst-case probability measure is a Gibbs probability measure and the unique solution to the maximization of the expected loss under a relative entropy constraint with respect to a reference probability measure. Fundamental generalization metrics, such as the sensitivity of the expected loss, the sensitivity of the empirical risk, and the generalization gap are shown to have closed-form expressions involving the worst-case data-generating probability measure. Existing results for the Gibbs algorithm, such as characterizing the generalization gap as a sum of mutual information and lautum information, up to a constant factor, are recovered. A novel parallel is established between the worst-case data-generating probability measure and the Gibbs algorithm. Specifically, the Gibbs probability measure is identified as a fundamental commonality of the model space and the data space for machine learning algorithms.

Mini-Batch Risk Forms

Risk forms are real functionals of two arguments: a bounded measurable function on a Polish space and a probability measure on that space. They are convenient mathematical structures adapting the coherent risk measures to the situation of a variable reference probability measure. We introduce a new class of risk forms called mini-batch forms. We construct them by using a random empirical probability measure as the second argument and by post-composition with the expected value operator. We prove that coherent and law invariant risk forms generate mini-batch risk forms which are well defined on the space of integrable random variables, and we derive their dual representation. We demonstrate how unbiased stochastic subgradients of such risk forms can be constructed. Then, we consider pre-compositions of mini-batch risk forms with nonsmooth and nonconvex functions, which are differentiable in a generalized way, and we derive generalized subgradients and unbiased stochastic subgradients of such compositions. Finally, we study the dependence of risk forms and mini-batch risk forms on perturbation of the probability measure and establish quantitative stability in terms of optimal transport metrics. We obtain finite-sample expected error estimates for mini-batch risk forms involving functions on a finite-dimensional space.

Efficient Multimodal Sampling via Tempered Distribution Flow

Sampling from high-dimensional distributions is a fundamental problem in statistical research and practice. However, great challenges emerge when the target density function is unnormalized and contains isolated modes. We tackle this difficulty by fitting an invertible transformation mapping, called a transport map, between a reference probability measure and the target distribution, so that sampling from the target distribution can be achieved by pushing forward a reference sample through the transport map. We theoretically analyze the limitations of existing transport-based sampling methods using the Wasserstein gradient flow theory, and propose a new method called TemperFlow that addresses the multimodality issue. TemperFlow adaptively learns a sequence of tempered distributions to progressively approach the target distribution, and we prove that it overcomes the limitations of existing methods. Various experiments demonstrate the superior performance of this novel sampler compared to traditional methods, and we show its applications in modern deep learning tasks such as image generation. The programming code for the numerical experiments is available in the supplementary material.

30 years of the ABFO study: Reproduction in a Brazilian sample

The ABFO study on third molar development is a benchmark in the scientific literature of dental age estimation. In its 30th anniversary, the study has been reproduced in the present external validation. Standardized comparative outcomes were obtained and discussed across studies. The sample consisted of 1.087 panoramic radiographs of Brazilian females (n=542, 49.87%) and males (n=545, 50.13%) between 14 and 22.9years. All available third molars were classified into developmental stages following Mincer's adaptation of Demirjian's system (8sequential stages, from A to H). The mean chronological age of individuals within each stage was assessed. The probability of an individual being≥18years was calculated for each third molar, sex and stage. Maxillary and mandibular third molars showed a similar development with an agreement between stages of about 90%. In general, males developed 0.5years (6months) earlier than females. The probability of being an adult increased considerably when at least one third molar is in stage G. Maxillary third molars had higher coefficients of determination (right: 0.704; left: 0.702), showing that statistical models with these teeth could explain better the age estimation outcomes. The reproducibility of the ABFO study on third molar development led to reference tables and probability measures for the studied Brazilian population.

Generative Stochastic Modeling of Strongly Nonlinear Flows with Non-Gaussian Statistics

Strongly nonlinear flows, which commonly arise in geophysical and engineering turbulence, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and analyze due to combination of high dimensionality and uncertainty, and there has been much interest in obtaining reduced models, in the form of stochastic closures, that can replicate their non-Gaussian statistics in many dimensions. Here, we propose a data-driven framework to model stationary chaotic dynamical systems through nonlinear transformations and a set of decoupled stochastic differential equations (SDEs). Specifically, we use optimal transport to find a transformation from the distribution of time-series data to a multiplicative reference probability measure such as the standard normal distribution. Then we find the set of decoupled SDEs that admit the reference measure as the invariant measure, and also closely match the spectrum of the transformed data. As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map. We demonstrate the application of this framework in Lorenz-96 system, a 10-dimensional model of high-Reynolds cavity flow, and reanalysis climate data. These examples show that SDE models generated by this framework can reproduce the non-Gaussian statistics of systems with moderate dimensions (e.g. 10 and more), and predict super-Gaussian tails that are not readily available from little training data. These findings suggest that this class of models provide an efficient hypothesis space for learning strongly nonlinear flows from small amounts of data.

Stationary Markov Nash Equilibria for Nonzero-Sum Constrained ARAT Markov Games

We consider a nonzero-sum Markov game on an abstract measurable state space with compact metric action spaces. The goal of each player is to maximize his respective discounted payoff function under the condition that some constraints on a discounted payoff are satisfied. We are interested in the existence of a Nash or noncooperative equilibrium. Under suitable conditions, which include absolute continuity of the transitions with respect to some reference probability measure, additivity of the payoffs and the transition probabilities (ARAT condition), and continuity in action of the payoff functions and the density function of the transitions of the system, we establish the existence of a constrained stationary Markov Nash equilibrium, that is, the existence of stationary Markov strategies for each of the players yielding an optimal profile within the class of all history-dependent profiles.

Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control

This paper deals with a nonlinear filtering problem in which a multi-dimensional signal process is additively affected by a process nu whose components have paths of bounded variation. The presence of the process nu prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satisfied by the unnormalized filtering process, and then we deduce the corresponding Kushner–Stratonovich equation. Under the condition that the jump times of the process nu do not accumulate over the considered time horizon, we show that the unnormalized filtering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable Itô-process.

Towards Event-Based State Estimation for Neuromorphic Event Cameras

In this work, a dynamic information extraction problem for neuromorphic event cameras is investigated from a state estimation perspective. The ego-motion pose estimation task of an event camera is formulated as a state estimation problem for a finite-state hidden Markov model subject to a special event-triggering mechanism. We model the threshold mismatch and the bandwidth limit of the event camera output generalization process as a stochastic event-triggering condition equipped with a state-dependent packet dropout process. For this problem, the recursive expression of the system state conditioned on the event-triggered measurement information is constructed under a suitably designed reference probability measure, based on which the event-based MMSE estimate for the considered estimation problem is further obtained. The effectiveness of proposed results is illustrated by numerical analysis and comparative evaluation of an ego-motion pose estimation example.

Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space . Each measure in the product is a shifted and scaled copy of a reference probability measure on that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

Valuing guaranteed minimum accumulation benefits by a change of numéraire approach

Three correlated risk factors, namely, financial, mortality and lapse risks, are modelled in an integrated way to support the valuation of guaranteed minimum accumulation benefits (GMAB). The dynamics of such risk factors under the risk-adjusted measure needed for valuation are derived. A sequence of reference probability measure changes is employed to obtain a compact and implementable price representations. We demonstrate how a change of numéraire approach facilitates the valuation of a GMAB rider in a variable annuity. A numerical illustration is presented to confirm the computational efficiency and accuracy of our proposed valuation methodology. In particular, our proposed approach on average takes only 0.07% of the computing time for the Monte-Carlo (MC) simulation technique to complete the valuation calculation. Furthermore, the standard errors of our approach's results are lower than those obtained from the MC-based computations. A sensitivity analysis is also performed with respect to various parameters. When there are no renewal options in a GMAB contract, we get the special case of a guaranteed minimum maturity benefit for which a closed-form valuation solution is established.

Event-triggered robust state estimation for systems with unknown exogenous inputs

An event-triggered robust state estimation problem for linear time-varying systems subject to Gaussian noises and non-stochastic unknown exogenous inputs is investigated in this work. To design the estimator, the state estimation problem is formulated as an optimization problem with a risk-sensitive cost function. This problem is solved by constructing a reference probability measure, under which the cost function has a simpler form and an information state can be developed. The obtained robust state estimator is shown to have a recursive form parameterized by a Riccati-type time-varying matrix equation. The effectiveness of the proposed event-based robust state estimator is illustrated with numerical examples.

Risk neutral reformulation approach to risk averse stochastic programming

The aim of this paper is to show that in some cases risk averse multistage stochastic programming problems can be reformulated in a form of risk neutral setting. This is achieved by a change of the reference probability measure making “bad” (extreme) scenarios more frequent. As a numerical example we demonstrate advantages of such change-of-measure approach applied to the Brazilian Interconnected Power System operation planning problem.

Stochastic analysis for measure-valued processes

In this paper, we introduce some recent progress on stochastic analysis for measure-valued processes, and propose some problems for further study in this direction. We first recall the notions of derivatives in measures, clarify their relations, and investigate functional inequalities of Dirichlet forms induced by these derivatives and reference probability measures. Then we construct diffusion processes on the Wasserstein space by using image dependent SDEs (stochastic differential equations), investigate the exponential ergodicity of the processes, and establish the Feynman-Kac formula for solutions of PDEs (partial differential equations) on the Wasserstein space. We also introduce Bismut formulas for the $L$ derivative of distribution dependent SDEs.

Event-Triggered Risk-Sensitive State Estimation for Hidden Markov Models

An event-triggered risk-sensitive state estimation problem for hidden Markov models is investigated in this work. The event-triggered scheme considered is fairly general, which covers most existing event-triggered conditions. By utilizing the reference probability measure approach, this estimation problem is reformulated as an equivalent one and solved. We show that the event-triggered risk-sensitive maximum a posteriori probability estimates can be obtained based on a newly defined unnormalized information state, which has a linear recursive form. Furthermore, the explicit solutions for two major classes of event-triggered conditions are derived if the measurement noise is Gaussian. A numerical comparison is provided to illustrate the effectiveness of the proposed results.

Efficient allocations under law-invariance: A unifying approach

We study the problem of optimising the aggregated utility within a system of agents under the assumption that individual utility assessments are law-invariant: they rank Savage acts merely in terms of their distribution under a fixed reference probability measure. We present a unifying framework in which optimisers can be found which are comonotone allocations of an aggregated quantity. Our approach can be localised to arbitrary rearrangement invariant commodity spaces containing at least all bounded wealths. The aggregation procedure is a substantial degree of freedom in our study. Depending on the choice of aggregation, the optimisers of the optimisation problems are allocations of a wealth with desirable economic efficiency properties, such as (weakly, biased weakly, and individually rationally) Pareto efficient allocations, core allocations, and systemically fair allocations.

The robust pricing–hedging duality for American options in discrete time financial markets

AbstractWe investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

Estimates for invariant probability measures of degenerate SPDEs with singular and path-dependent drifts

In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic (partial) differential equations are proved to possess the weak existence and uniqueness of solutions, as well as the existence, uniqueness and entropy estimates of invariant probability measures. When the reference measure satisfies the log-Sobolev inequality, Sobolev estimates are derived for the density of invariant probability measures. Some results are new even for non-degenerate SDEs with path-independent drifts. The main results are applied to nonlinear functional SPDEs and degenerate functional SDEs/SPDEs.

Kyle--Back Equilibrium Models and Linear Conditional Mean-Field SDEs

In this paper we study the Kyle--Back strategic insider trading equilibrium model in which the insider has instantaneous information on an asset, assumed to follow an Ornstein--Uhlenback-type dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between an insider's information and the market price, and it is the first time being put into a rigorous mathematical framework of the recently developed conditional mean-field stochastic differential equation (CMFSDE). With the help of the “reference probability measure" concept in filtering theory, we shall first prove a general well-posedness result for a class of linear CMFSDEs, which is new in the literature of both filtering theory and mean-field SDEs and will be the foundation for the underlying strategic equilibrium model. Assuming some further Gaussian structures of the model, we find a closed form of optimal intensity of trading strategy as well as the dynamic pricing rules, and we substantiate the well-posedness of the resulting optimal closed-loop system, whence the existence of Kyle--Back equilibrium.

General dynamic term structures under default risk

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath et al. (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.