Properties Of Conditional Expectation Research Articles

Novel Criteria of Stochastic Stability for Discrete-Time Markovian Jump Singular Systems via Supermartingale Approach

This article deals with the problem of stochastic stability for a class of discrete-time Markovian jump singular systems. A time-dependent coordinate transformation is provided, under which an equivalent form of the original Markovian jump singular systems can be obtained. This equivalent form not only shows the inherent state jump behavior at the switching instants, but also plays an important role in the stochastic stability analysis. Constructing a supermartingale over some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -algebra together with the property of conditional expectation, a necessary and sufficient condition is established to ensure that the equilibrium point of the underlying system is exponentially stable in mean square. Two numerical examples are given to illustrate the effectiveness of the developed results.

H ∞ filtering of discrete-time Markovian jump singular systems via bounded real lemma and supermartingale-liked approach

This paper addresses the filtering for discrete-time Markovian jump singular systems (MJSSs). While the existing researches have only given the sufficient conditions, in this paper a sufficient and necessary condition is established to guarantee that the filtering error system is regular, causal, stochastically stable and performance in light of supermartingale-liked approach and property of conditional expectation as well as – bounded real lemma. For obtaining the filter, an equivalent matrix inequality to the obtained one in nonlinear form is provided in terms of congruence transformation. Thus the desired filter can be developed by choosing matrix variables with certain structures. Finally, two simulations are provided to show the effectiveness of the proposed techniques in this paper.

A note on conditional expectation for Markov kernels

A known property of conditional expectation is extended to the framework of Markov kernels. Its meaning in terms of densities is provided. Some examples located in the field of clinical diagnosis are presented to delimit the main result of the paper.

Integrating macroeconomic variables into behavioral models for interest rate risk measurement in the banking book

Recent Basel Committee on Banking Supervision standards on interest rate risk in the banking book require the consideration of macroeconomic variables for modeling client behaviors, while no macroeconomic risk scenarios are prescribed by regulators or are generally agreed in the industry. Since macroeconomic variables and interest rates are correlated, projecting macroeconomic variables for interest rate risk measurement poses the challenge of maintaining consistency with regulator-prescribed interest rate scenarios. This paper proposes an approach to integrate macroeconomic variables with interest rate scenarios. The conditional expectation of macroeconomic variables on interest rate variables is used to capture their interdependence. Based on the mathematical properties of conditional expectation, we derive its nonparametric estimator. The resulting projections of macroeconomic variables are fully consistent with the given interest rate scenarios and are convenient for implementation in practice. An empirical application to Canadian fixed-term deposits is conducted to illustrate the proposed approach.

The Pricing of Total Return Swap Under Default Contagion Models with Jump-Diffusion Interest Rate Risk

In this paper, we consider a two-firm default contagion model with counterparty risk and jumpdiffusion interest rate risk. Under this model, we study the pricing of total return swap(TRS). We assume that the interest rate follows the Vasicek jump-diffusion model, and obtain the Libor market interest rate. The case that default is related to the interest rate is considered. Using the method of change of measure and the properties of conditional expectations, the joint conditional survival probability and joint conditional density function are derived. Applying the arbitragefree pricing principle of TRS in the complete market, we price the swap rate of TRS and obtain the closed-form solution. And we analyze the effect of various factors on the price of TRS.

Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals

The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.

Characterizations based on regression assumptions of order statistics

Let X(1)≤X(2)≤···≤X(n) be the order statistics from independent and identically distributed random variables {Xi, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.

A novel suboptimal algorithm for state estimation of Markov jump linear systems

This paper is concerned with state estimation problem for Markov jump linear systems where the disturbances involved in the systems equations and measurement equations are assumed to be Gaussian noise sequences. Based on two properties of conditional expectation, orthogonal projective theorem is applied to the state estimation problem of the considered systems so that a novel suboptimal algorithm is obtained. The novelty of the algorithm lies in using orthogonal projective theorem instead of Kalman filters to estimate the state. A numerical comparison of the algorithm with the interacting multiple model algorithm is given to illustrate the effectiveness of the proposed algorithm.

Separating Sets and Conditional Expectation

In the parlance of modern probability theory, collections of functions sufficiently rich to distinguish probability measures are called separating sets. Despite being very elementary in nature, separating sets are only encountered in advanced study of Markov processes. In this note we show how they can be used on an elementary level for facilitating proofs of some interesting properties of conditional expectation.

Conditional Expectations for Unbounded Operator Algebras

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of anO∗-algebra into the Hilbert space on which theO∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear mapℰof anO∗-algebraℳonto a givenO∗-subalgebra𝒩ofℳ. Here the domainD(ℰ)ofℰdoes not equal toℳin general, and so such a conditional expectation is called unbounded.

Upper Bounds for Ruin Probability with Stochastic Investment Return

Risk models with stochastic investment return are widely held in practice, as well as in more challenging research fields. Risk theory is mainly concerned with ruin probability, and a tight bound for ruin probability is the best for practical use. This paper presents a discrete time risk model with stochastic investment return. Conditional expectation properties and martingale inequalities are used to obtain both exponential and non-exponential upper bounds for the ruin probability.

Compact convergence of σ-fields and relaxed conditional expectation

The collection of sub-σ-fields of a Borel measure space when endowed with the topology of strong convergence is in general not a compact space. The paper offers a completion of this space which makes it compact. The elements which are added to the space are called relaxed σ-fields. A notion of relaxed conditional expectation with respect to a relaxed σ-field is identified. The relaxed conditional expectation is a probability measure-valued map. It is shown that the conditional expectation operator is continuous on the completion of the space. Other properties of conditional expectation are lifted to and interpreted in the relaxed framework.

Measurable majorants in L1

Given a probability space (X, ℱ, μ) and a σ-algebra A ⊂ ℱ, arguably the most powerful tool in gaining information about an ℱ-measurable function f from restricted knowledge of -measurability is that of the conditional expectation E(f | ); written throughout the remainder of this note. Two properties of conditional expectation that may be exploited to gain information, but which also limit conditional expectation's use are the following.

On the existence of quantum subdynamics

It is shown, using only elementary operator algebra, that an open quantum system coupled to its environment will have a subdynamics (reduced dynamics) as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated. Furthermore, it is proved that for a finite temperature the KMS condition for the lowest-order correlation function cannot be reproduced by any type of linear subdynamics except the reversible Hamiltonian one of a closed system. The first statement can be seen as a particular case of a more general theorem of Takesaki on the properties of conditional expectations in von Neumann algebras. The concept of subdynamics used here allows for memory effects, no assumption is made of a Markov property. For dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem.

Left and right linear innovations for a multivariate SαS random variable

Properties of conditional expectations and metric projections for multivariate symmetric α-stable random variables with 1 < α < 2 are studied.

Optimal Linear Filtering and Smoothing for a Discrete Time Stable Linear Model

Properties of conditional expectation and metric projection for multivariate symmetric stable distributions are studied. Linear filtering, smoothing, and prediction problems for a discrete time stable linear model with constant coefficients are solved.

Some special properties of conditional expectation

Some special properties of conditional expectation

Properties of conditional expectations and variances of a Wiener process observed in the presence of noise

Properties of conditional expectations and variances of a Wiener process observed in the presence of noise

Stochastic approximation with correlated data

New almost sure convergence results for a special form of the multidimensional Robbins-Monro stochastic approximation procedure are developed. The results are applicable to cases where the training is heavily correlated. No conditional expectation properties or boundedness assumptions are required to apply the new results. For example, when the data sequence is normal and i) M -dependent, ii) autoregressive moving average, or iii) band-limited, the results can be used to establish the almost sure convergence of each algorithm treated. The special form of the Robbins-Monro procedure considered is motivated by a consideration of several algorithms that have been proposed for discrete-time adaptive signal-processing applications. Most of these algorithms can also be viewed as stochastic gradient-following algorithm. The ease with which the new results can be applied is illustrated.

Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, D the real Schwartz space, and D′ its strong dual. D and D′ are partially ordered by C and C′ respectively, where C is the positive cone of nonnegative functions in D and C′ its dual in D′ . C is a strict B -cone and C′ is normal, where B is the family of all bounded subsets of D . If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y( ω) − X( ω) ∈ D′ for almost all ω ∈ Ω(P). Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.