Conditional Expectation Operators Research Articles

Strong completeness of a class of 𝐿²(𝑇)-type Riesz spaces

The L p , 1 ≤ p ≤ ∞ L^{p}, 1\le p\le \infty , spaces have been generalized to the setting of Riesz spaces as L p ( T ) {L}^{p}(T) spaces, on which there are R ( T ) R(T) -valued norms. The strong sequential completeness of the space L 1 ( T ) {L}^{1}(T) and the strong completeness of L ∞ ( T ) {L}^{\infty }(T) with resepct to their respective R ( T ) R(T) -valued norms were established by Kuo, Rodda, and Watson. In the current work, the T T -strong completeness of L 2 ( T ) {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T T is a weak order unit for the T T -strong dual.

Generating functions in Riesz spaces

We introduce and study the concept of generating function for natural elements in a Dedekind complete Riesz space equipped with a conditional expectation operator. This allows us to study discrete processes in a free-measure setting. In particular we improve a result obtained by Kuo, Vardy and Watson concerning Poisson approximation.

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [B.A. Watson, An Andô-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558] and a Riesz-Frechet representation theorem for the T-strong dual, where T is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required.

Measuring Information From Moments

We investigate the problem of representing information measures in terms of the moments of the underlying random variables. First, we derive polynomial approximations of the conditional expectation operator. We then apply these approximations to bound the best mean-square error achieved by a polynomial estimator—referred to here as the PMMSE. In Gaussian channels, the PMMSE coincides with the minimum mean-square error (MMSE) if and only if the input is either Gaussian or constant, i.e., if and only if the conditional expectation of the input of the channel given the output is a polynomial of degree at most 1. By combining the PMMSE with the I-MMSE relationship, we derive new formulas for information measures (e.g., differential entropy, mutual information) that are given in terms of the moments of the underlying random variables. As an application, we introduce estimators for information measures from data via approximating the moments in our formulas by sample moments. These estimators are shown to be asymptotically consistent and possess desirable properties, e.g., invariance to affine transformations when used to estimate mutual information.

Conditional supremum in Riesz spaces and applications

We extend the concept of conditional supremum to the measure-free setting of Riesz spaces via the conditional expectation operator. We explore its properties and show how this tool is crucial in generalizing various results across multiple disciplines to the framework of Riesz spaces. Among other applications, we utilize this concept in finance to derive characterizations of certain financial conditions.

Dynamic bid–ask pricing under Dempster-Shafer uncertainty

We deal with the problem of pricing in a multi-period binomial market model, allowing for frictions in the form of bid–ask spreads. We introduce and characterize time-homogeneous Markov multiplicative binomial processes under Dempster-Shafer uncertainty together with the induced conditional Choquet expectation operator. Given a market formed by a frictionless risk-free bond and a non-dividend paying stock with frictions, we prove the existence of an equivalent one-step Choquet martingale belief function. We then propose a dynamic Choquet pricing rule with bid–ask spreads showing that the discounted lower price process of a European derivative contract on the stock is a Choquet super-martingale. We finally provide a normative justification in terms of a dynamic generalized no-arbitrage condition relying on the notion of partially resolving uncertainty due to Jaffray.

Meta Derivative Identity for the Conditional Expectation

Consider a pair of random vectors (X,Y) and the conditional expectation operator E[X|Y = y]. This work studies analytical properties of the conditional expectation by characterizing various derivative identities. The paper consists of two parts. In the first part of the paper, a general derivative identity for the conditional expectation is derived. Specifically, for the Markov chain U ↔ X ↔ Y, a compact expression for the Jacobian matrix of E[ψ(Y,U)|Y = y] for a smooth function ψ is derived. In the second part of the paper, the main identity is specialized to the exponential family and two main applications are shown. First, it is demonstrated that, via various choices of the random vector U and function ψ, one can recover and generalize several known identities (e.g., Tweedie’s formula) and derive some new ones. For example, a new relationship between conditional expectations and conditional cumulants is established. Second, it is demonstrated how the derivative identities can be used to establish new lower bounds on the estimation error. More specifically, using one of the derivative identities in conjunction with a Poincaré inequality, a new lower bound on the minimum mean squared error, which holds for all prior distributions on the input signal, is derived. The new lower bound is shown to be tight in the high-noise regime for the additive Gaussian noise setting.

Sharp estimates for conditionally centered moments and for compact operators on Lp$L^p$ spaces

AbstractLet be a probability space, ξ be a random variable on , be a sub‐σ‐algebra of , and let be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of in terms of the moments of ξ. This allows us to find the optimal constant in the bounded compact approximation property of , .

On the Composition of Conditional Expectation and Multiplication Operators

In this paper, first we provide some necessary and sufficient conditions for quasi- normality and quasi- hyponormality of weighted conditional type operators. And then the spectrum, residual spectrum, point spectrum and spectral radius of weighted conditional type operators are computed. As an application, we give an equivalent conditions for weighted conditional type operators to be quasinilpotent. Also, some examples are provided to illustrate concrete applications of the main results.

A general linear quadratic stochastic control and information value

This paper introduces a generalized linear quadratic problem (for short, GLQ) in which two cooperative controllers are posed with two types of constraints both: admissibility (input) constraint and information (randomness) constraint. Our GLQ includes most existing stochastic LQ forms as its special cases, and also widens new frontier beyond. We first derive a generalized stochastic maximum principle for GLQ which involves projection and conditional expectation operators both. Then, focusing on information constraint, we further study its open-loop solvability in dynamically recursive context. Moreover, we derive and compare the optimal values for various information structures of GLQ via closed-loop representation, that reveals a novel insight on underlying information capacity. In particular, information redundancy, efficiency and monotonicity in stochastic LQ setting are first addressed.

Stochastic algorithmic differentiation of (expectations of) discontinuous functions (indicator functions)

In this paper, we present a method for the accurate estimation of the derivative (aka. sensitivity) of expectations of functions involving an indicator function by modifying a (stochastic) algorithmic differentiation, replacing the derivative of the indicator function with a suitable operator. We show that we can split this operator into a conditional expectation operator and the density. This allows using different or improved numerical approximation methods for these operators, e.g. regression. The method is an improvement of the approach presented in C.P. Fries [Automatic backward differentiation for American Monte-Carlo algorithms (conditional expectation). Risk, April, 2018; Stochastic automatic differentiation: Automatic differentiation for Monte-Carlo simulations, Quant. Finance 19(6) (2019), pp. 1043–1059]. The finite difference approximation of a partial derivative of a Monte-Carlo integral of a discontinuous function is known to exhibit a high Monte-Carlo error. The issue is evident since the Monte-Carlo approximation of a discontinuous function is just a finite sum of discontinuous functions and as such, not even differentiable. The algorithmic differentiation of a discontinuous function is problematic. A natural approach is to replace the discontinuity by continuous functions. This is equivalent to replacing a path-wise automatic differentiation by a (local) finite difference approximation. The decoupling of the integration of the Dirac delta and the remaining conditional expectation introduced here results in an improvement in terms of variance reduction and implementation design.

Substitution vector-valued integral type operators on Orlicz spaces

In this paper, we consider some fundamental properties of a substitution vector-valued integral operator Tu? from Orlicz space L?(?) to Hilbert space H by the language of conditional expectation operators. First, we present necessary and sufficient conditions for boundedness and compactness Tu? from L?(?) to H. Next, we investigate the problem of conditions on the generating Young functions, the functions u, ? and h = d(? ? ?-1)/d?, under which operator Tu? is of closed range or finite rank. Finally, we determine the lower and upper estimates for the essential norm of Tu? on Orlicz spaces under certain conditions.

Doob’s maximal inequalities for martingales in variable Lebesgue space

In this paper we deal with the martingales in variable Lebesgue space over a probability space. We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space. The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent. In particular, we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities. Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p > 1.

On (in)dependence measures in Riesz spaces

In this paper we introduce an independence measure between a vector f in a Riesz space E and a conditional expectation operator T on E. We show that this measure satisfies some postulates corresponding to these formulated by Renyi (1959) for dependence measure between random variables. The resulting dependence measure in E is also investigated. Some examples of application are given.

Products of Conditional Expectation Operators: Convergence and Divergence

In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $(\Omega,\cal{F},P)$ is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-$\sigma$-fields of $\cal{F}$ can be constructed for a large class of random variables in $L^2(\Omega,\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\Omega,\cal{F},P)$ is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub-$\sigma$-fields of $\cal{F}$ must converge for all random variables in $L^1(\Omega,\cal{F},P)$.

Viewing *-multiplication operators between Orlicz spaces

In this paper, Lambert multipliers acting between Orlicz spaces are characterized based on some properties of conditional expectation operators. We provide a necessary and sufficient condition for the *-multiplication operators to have closed range. Finally, a necessary condition for Fredholmness of these type of operators will be investigated.

On a class of bounded operators and their ascent and descent

In this paper we provide some conditions under which the ascent and descent of the weighted conditional expectation operators of the form of $$M_wEM_u$$ on $$L^p$$ -spaces are finite. Moreover, we give some necessary and sufficient conditions for $$M_wEM_u$$ to be power bounded. In the sequel we apply some results in operator theory on ascent and descent to $$M_wEM_u$$ . Finally we find that $$T=M_wEM_u$$ is Cesaro bounded if and only if $${\widehat{T}}$$ is Cesaro bounded.

A Regression Approach to Certain Information Transmission Problems

A general information transmission model, under independent and identically distributed Gaussian codebook and nearest neighbor decoding rule with processed channel output, is investigated using the performance metric of generalized mutual information. When the encoder and the decoder know the statistical channel model, it is found that the optimal channel output processing function is the conditional expectation operator, thus hinting a potential role of regression, a classical topic in machine learning, for this model. Without utilizing the statistical channel model, a problem formulation inspired by machine learning principles is established, with suitable performance metrics introduced. A data-driven inference algorithm is proposed to solve the problem, and the effectiveness of the algorithm is validated via numerical experiments. Extensions to more general information transmission models are also discussed.

On the dilation of a conditional operator

ABSTRACT Let E be the conditional expectation operator with respect to a σ-finite subalgebra of Σ. This paper is concerned with the study of a conditional type operator on of the form , where u and v are measurable functions and Q=I−E. We discuss matrix theoretic characterizations of several properties of with an application-oriented approach.

Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.