scholarly journals Inclusions of ternary rings of operators and conditional expectations

2013 ◽  
Vol 155 (3) ◽  
pp. 475-482 ◽  
Author(s):  
PEKKA SALMI ◽  
ADAM SKALSKI
Keyword(s):  
Conditional Expectation ◽  
Ternary Ring ◽  
Conditional Expectations

AbstractIt is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P: T → X then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.

scholarly journals Tail Conditional Expectations for Exponential Dispersion Models

2005 ◽  
Vol 35 (1) ◽  
pp. 189-209 ◽  
Author(s):  
Zinoviy Landsman ◽  
Emiliano A. Valdez
Keyword(s):  
Conditional Expectation ◽  
Value At Risk ◽  
Risk Measure ◽  
Inverse Gaussian ◽  
Dispersion Models ◽  
Average Amount ◽  
Tail Conditional Expectation ◽  
Conditional Expectations ◽  
Exponential Dispersion Models ◽  
Fixed Period

There is a growing interest in the use of the tail conditional expectation as a measure of risk. For an institution faced with a random loss, the tail conditional expectation represents the conditional average amount of loss that can be incurred in a fixed period, given that the loss exceeds a specified value. This value is typically based on the quantile of the loss distribution, the so-called value-at-risk. The tail conditional expectation can therefore provide a measure of the amount of capital needed due to exposure to loss. This paper examines this risk measure for “exponential dispersion models”, a wide and popular class of distributions to actuaries which, on one hand, generalizes the Normal and shares some of its many important properties, but on the other hand, contains many distributions of nonnegative random variables like the Gamma and the Inverse Gaussian.

scholarly journals Positive Dependence of Signals

2010 ◽  
Vol 47 (3) ◽  
pp. 893-897 ◽  
Author(s):  
Michel Denuit
Keyword(s):  
Conditional Expectation ◽  
Conditional Variance ◽  
Sufficient Condition ◽  
Convex Order ◽  
Positive Dependence ◽  
Increasing Convex Order ◽  
Quantity Of Interest ◽  
Conditional Expectations ◽  
Special Case

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

BAYES' FORMULA FOR SECOND QUANTIZATION OPERATORS

2006 ◽  
Vol 06 (02) ◽  
pp. 245-253 ◽  
Author(s):  
ALBERTO LANCONELLI
Keyword(s):  
Conditional Expectation ◽  
Bayes Rule ◽  
Second Quantization ◽  
Absolutely Continuous ◽  
Bayes Formula ◽  
Conditional Expectations ◽  
Quantization Operator ◽  
Continuous Measures ◽  
The Relationship ◽  
Absolutely Continuous Measures

The Bayes' formula provides the relationship between conditional expectations with respect to absolutely continuous measures. The conditional expectation is in the context of the Wiener space — an example of second quantization operator. In this note we obtain a formula that generalizes the above-mentioned Bayes' rule to general second quantization operators.

scholarly journals Conditional expectation for operator-valued measures and functions

1984 ◽  
Vol 30 (3) ◽  
pp. 421-429 ◽  
Author(s):  
Brian Jefferies
Keyword(s):  
Conditional Expectation ◽  
Existence And Uniqueness ◽  
Conditional Expectations ◽  
Nikodym Theorem

A Radon–Nikodým theorem for operator-valued measures is applied to deduce the existence and uniqueness of conditional expectations in certain cases.

scholarly journals SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

2011 ◽  
Vol 22 (07) ◽  
pp. 1031-1050
Author(s):  
ESTEBAN ANDRUCHOW ◽  
GABRIEL LAROTONDA
Keyword(s):  
Conditional Expectation ◽  
Von Neumann Algebra ◽  
Boundedness Condition ◽  
Derivative Operator ◽  
Von Neumann ◽  
Conditional Expectations ◽  
Finite Trace

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.

GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX

1994 ◽  
Vol 05 (02) ◽  
pp. 169-178 ◽  
Author(s):  
ESTEBAN ANDRUCHOW ◽  
DEMETRIO STOJANOFF
Keyword(s):  
Homogeneous Space ◽  
Finite Index ◽  
Conditional Expectation ◽  
Von Neumann Algebras ◽  
Covering Space ◽  
Von Neumann ◽  
Similarity Orbit ◽  
Conditional Expectations

Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [Formula: see text] are von Neumann algebras. We prove that the similarity orbit of e by invertibles of [Formula: see text] is an homogeneous space iff the index of E is finite. If also [Formula: see text], then this orbit is a covering space for the orbit of E.

scholarly journals A relation between positive dependence of signal and the variability of conditional expectation given signal

2006 ◽  
Vol 43 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Toshihide Mizuno
Keyword(s):  
Conditional Expectation ◽  
Marginal Distribution ◽  
Stochastic Order ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Distributions ◽  
Convex Order ◽  
Positive Dependence ◽  
Dispersive Order ◽  
Conditional Expectations

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

scholarly journals Tail Conditional Expectations for Exponential Dispersion Models

2005 ◽  
Vol 35 (01) ◽  
pp. 189-209 ◽  
Author(s):  
Zinoviy Landsman ◽  
Emiliano A. Valdez
Keyword(s):  
Conditional Expectation ◽  
Value At Risk ◽  
Risk Measure ◽  
Inverse Gaussian ◽  
Dispersion Models ◽  
Average Amount ◽  
Tail Conditional Expectation ◽  
Conditional Expectations ◽  
Exponential Dispersion Models ◽  
Fixed Period

There is a growing interest in the use of the tail conditional expectation as a measure of risk. For an institution faced with a random loss, the tail conditional expectation represents the conditional average amount of loss that can be incurred in a fixed period, given that the loss exceeds a specified value. This value is typically based on the quantile of the loss distribution, the so-called value-at-risk. The tail conditional expectation can therefore provide a measure of the amount of capital needed due to exposure to loss. This paper examines this risk measure for “exponential dispersion models”, a wide and popular class of distributions to actuaries which, on one hand, generalizes the Normal and shares some of its many important properties, but on the other hand, contains many distributions of nonnegative random variables like the Gamma and the Inverse Gaussian.

Larotonda spaces: Homogeneous spaces and conditional expectations

2016 ◽  
Vol 27 (02) ◽  
pp. 1650002
Author(s):  
Esteban Andruchow ◽  
Lázaro Recht
Keyword(s):  
Conditional Expectation ◽  
Quotient Space ◽  
Homogeneous Spaces ◽  
Tangent Vector ◽  
Linear Connection ◽  
Unitary Groups ◽  
Finsler Metric ◽  
Differentiable Structure ◽  
Conditional Expectations ◽  
Vector Formula

We define a Larotonda space as a quotient space [Formula: see text] of the unitary groups of [Formula: see text]-algebras [Formula: see text] with a faithful unital conditional expectation [Formula: see text] In particular, [Formula: see text] is complemented in [Formula: see text], a fact which implies that [Formula: see text] has [Formula: see text] differentiable structure, with the topology induced by the norm of [Formula: see text]. The conditional expectation also allows one to define a reductive structure (in particular, a linear connection) and a [Formula: see text]-invariant Finsler metric in [Formula: see text]. Given a point [Formula: see text] and a tangent vector [Formula: see text], we consider the problem of whether the geodesic [Formula: see text] of the linear connection satisfying these initial data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.

scholarly journals Positive Dependence of Signals

2010 ◽  
Vol 47 (03) ◽  
pp. 893-897 ◽  
Author(s):  
Michel Denuit
Keyword(s):  
Conditional Expectation ◽  
Conditional Variance ◽  
Sufficient Condition ◽  
Convex Order ◽  
Positive Dependence ◽  
Increasing Convex Order ◽  
Quantity Of Interest ◽  
Conditional Expectations ◽  
Special Case

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L 2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

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