Integrals Of Processes Research Articles

An innovative blockchain-based system for human resources digitalization with traceable relationship management

Small and Midsize Enterprises (SMEs) have faced innovative changes in their organizational structure, especially in human resources (HR) and employee management: these digital structural changes have transformed their business practices and strategies. The employees with new technical skills have also been impactful in this digitalization. The HR departments in SMEs have become increasingly pivotal in recruiting new employees and supporting existing employees throughout the organization. ITSvil in Salerno (Italy) has developed the Easywork project based on the “Fenice” microservices architecture. This system supports the HR departments in efficiently managing HR activities through digitalization. The first step in acquiring new resources is formulating the employment contract, which is essential for stipulating agreements on specific job requirements. Blockchain technology and smart contracts are among recent technological innovations impacting this process. Writing contracts in digital codes to simplify and effectively manage the legal relationship involving bureaucratic tasks and cost management is now possible. The new digital system makes employment contract processing transparent, traceable, and reliable. This research presents the design and development of a new microservice compatible and integrable process with the “Fenice” system based on blockchain technology. The proposed digital system aims to make HR management activities more transparent and traceable.

Symmetric Fuzzy Stochastic Differential Equations Driven by Fractional Brownian Motion

We consider symmetric fuzzy stochastic differential equations where diffusion and drift terms arise in a symmetric way at both sides of the equations and diffusion parts are driven by fractional Brownian motions. Such equations can be used in real-life hybrid systems, which include properties of being both random and fuzzy and reflecting long-range dependence. By imposing on the mappings occurring in the equation the conditions of Lipschitzian continuity and additional constraints by an integrable stochastic process, we construct an approximation sequence of fuzzy stochastic processes and apply this to prove the existence of a unique solution to the studied equation. Finally, a model from population dynamics is considered to illustrate the potential application of our equations.

On estimation and prediction in spatial functional linear regression model

We consider a spatial functional linear regression, where a scalar response is related to a square integrable spatial functional process. We use a smoothing spline estimator for the functional slope parameter and establish a finite sample bound for variance of this estimator under mixing spatial dependence. Then, we give a bound of the prediction error. Finally, we illustrate our results by simulations

A Monte Carlo algorithm for multiple stochastic integrals of stable processes

In this paper, we provide a Monte Carlo method to calculate multiple stochastic integrals which is based on a.s. distributional convergence.

Optimal stopping under g-Expectation with -integrable reward process

AbstractIn this paper we study a class of optimal stopping problems under g-expectation, that is, the cost function is described by the solution of backward stochastic differential equations (BSDEs). Primarily, we assume that the reward process is $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable with $\mu>\mu_0$ for some critical value $\mu_0$ . This integrability is weaker than $L^p$ -integrability for any $p>1$ , so it covers a comparatively wide class of optimal stopping problems. To reach our goal, we introduce a class of reflected backward stochastic differential equations (RBSDEs) with $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable parameters. We prove the existence, uniqueness, and comparison theorem for these RBSDEs under Lipschitz-type assumptions on the coefficients. This allows us to characterize the value function of our optimal stopping problem as the unique solution of such RBSDEs.

Factorisation in stopping-time Banach spaces: Identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.

Deep Gaussian processes and infinite neural networks for the analysis of EEG signals in Alzheimer’s diseases

Deep neural network models (DGPs) can be represented hierarchically by a sequential composition of layers. When the prior distribution over the weights and biases are independently identically distributed, there is an equivalence with Gaussian processes (GP) in the limit of an infinite net[1]work width. DGPs are non-parametric statistical models used to character[1]ize patterns of complex non-linear systems due to their flexibility, greater generalization capacity, and a natural way of making inferences about the parameters and states of the system. This article proposes a hierarchi[1]cal Bayesian structure to model the weights and biases of a deep neural network. We deduce a general formula to calculate the integrals of Gaussian processes with non-linear transfer densities and obtain a kernel to estimate the covariance functions. In the methodology, we conduct an empirical study analyzing an electroencephalogram (EEG) database for diagnosing Alzheimer’s disease. Additionally, the DGPs models are esti[1]mated and compared with the NN models for 5, 10, 50, 100, 500, and 1000 neurons in the hidden layer, considering two transfer functions: Recti[1]fied Linear Unit (ReLU) and hyperbolic Tangent (Tanh). The results show good performance in the classification of the signals. Finally, we use the mean square error as a goodness of fit measure to validate the proposed models, obtaining low estimation errors.

Skorohod and Stratonovich integrals for controlled processes

Given a continuous Gaussian process x which gives rise to a p-geometric rough path for p∈(2,3), and a general continuous process y controlled by x, under proper conditions we establish the relationship between the Skorohod integral ∫0tysd♢xs and the Stratonovich integral ∫0tysdxs. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.

Two-loop hexa-box integrals for non-planar five-point one-mass processes

We present the calculation of the three distinct non-planar hexa-box topologies for five-point one-mass processes. These three topologies are required to obtain the two-loop virtual QCD corrections for two-jet-associated W, Z or Higgs-boson production. Each topology is solved by obtaining a pure basis of master integrals and efficiently constructing the associated differential equation with numerical sampling and unitarity-cut techniques. We present compact expressions for the alphabet of these non-planar integrals, and discuss some properties of their symbol. Notably, we observe that the extended Steinmann relations are in general not satisfied. Finally, we solve the differential equations in terms of generalized power series and provide high-precision values in different regions of phase space which can be used as boundary conditions for subsequent evaluations.

Sensitivity of option prices via fuzzy Malliavin calculus

In this paper, we define the Malliavin derivative and the Skorohod integral for fuzzy stochastic processes for the sensitivity analysis of the option prices. The price dynamics could be modelled by equations involving uncertainties that lead to modelling with fuzzy processes in equations. We introduce a fuzzy stochastic differential equation for financial pricing models under an asset price that follows a fuzzy stochastic process. If there is no explicit solution to the equations, the sensitivity analysis, and the estimates are computationally expensive, and the result will be inaccurate due to the estimates and the derivative of the payoff function. The Malliavin calculus and the integration by parts formula in fuzzy space can be exploited to bypass the derivative of the payoff function.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $H>1/2$, and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

On Decoupling in Banach Spaces

We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–Gundy-type inequalities for stochastic integrals of X-valued processes can be obtained from decoupling inequalities for X-valued dyadic martingales.

Quantum Hermite-Hadamard type integral inequalities for convex stochastic processes

<abstract><p>In this paper, we introduce the notions of $ q $-mean square integral for stochastic processes and co-ordinated stochastic processes. Furthermore, we establish some new quantum Hermite-Hadamard type inequalities for convex stochastic processes and co-ordinated stochastic processes via newly defined integrals. It is also revealed that the results presented in this research transformed into some already proved results by considering the limits as $ q, \; q_{1}, \; q_{2}\rightarrow 1^{-} $ in the newly obtained results.</p></abstract>

Some generalised integral inequalities for bidimensional preinvex stochastic processes

In this study, we generalized some integral inequalities for bidimensional preinvex stochastic processes. For this reason, we used mean-square integrable preinvex stochastic processes on the real line and on the coordinates, respectively. Therefore, we obtained some generalized integral inequalitİies for preinvex stochastic processes on the real line.

The convergence and stability of full discretization scheme for stochastic age-structured population models

In this paper, a fully discretization scheme based on the implicit Euler method (IM) is considered for stochastic age-structured population models. The preservation of the total population with a suitable numerical boundary condition according to the biological meanings are shown. An explicit formula of the numerical basic reproductive number Rh is proposed by the technique that the numerical process is embedded into an l1(R)-valued integrable stochastic process with infinite stochastic Leslie operators. Furthermore, the convergence and connection between Rh and the stability of numerical solution is analyzed. The preservation and detection of the analytic stability through the numerical solutions are discussed for small stepsize. Finally, some numerical experiments including an infection-age model for modified SARS epidemic illustrate the verification and efficiency of our analysis.

Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

Abstract A generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.

Two-loop integrals for planar five-point one-mass processes

We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in ϵ. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

On optimal recovery of integrals of random processes

In this paper we prove a sharp Ostrowski type inequality for random processes of certain classes. This inequality is later applied to a solution of the optimal recovery of the integral ∫01ξtdt, using the random variables ξτ1,…,ξτn as an information set, where τ1,…,τn are random variables. We also consider the problem of the information set optimization.

Exponential functionals of Markov additive processes

We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Lévy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.

𝕃p solutions of reflected backward stochastic differential equations with jumps

Given p ∈ (1, 2), we study 𝕃p-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator g is Lipschitz continuous in (y, z, u). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with p-integrable parameters admits a unique 𝕃p solution via a fixed-point argument. The Y -component of the unique 𝕃p solution can be viewed as the Snell envelope of the reflecting obstacle 𝔏 under g-evaluations, and the first time Y meets 𝔏 is an optimal stopping time for maximizing the g-evaluation of reward 𝔏.