Journal articles on the topic 'G-stochastic integral'

In this paper is introduced the concept of (m, h1, h2)-convexity for stochastic processes dominated by other stochastic processes with the same property, some mean square integral Hermite-Hadamard type inequalities for this kind of generalized convexity are established and from the founded results, other mean square integral inequalities for the classical convex, s-convex in the first and second sense, P-convex and MT-convex stochastic processes are deduced.

The jump-diffusion stochastic process is one of the most common forms in reality (such as wave propagation, noise propagation, turbulent flow, etc.), and researchers often refer to them in models of random processes such as Wiener process, Levy process, Ito-Hermite process, in research of G. D. Nunno, B. Oksendal, F. B. Hanson, etc. In our research, we have reviewed and solved three problems: (1) Jump-diffusion process (also known as the Ito-Levy process); (2) Solve the differential equation jump-diffusion random linear, in the case of one-dimensional; (3) Calculate the Wiener-Ito integral to the random Ito-Hermite process. The main method for dealing with the problems in our presentation is the Ito random-integrable mathematical operations for the continuous random process associated with the arbitrary differential jump by the Poisson random measure. This study aims to analyse the basic properties of jump-diffusion process that are solutions to the jump-diffusion linear stochastic differential equations: dX(t) = [a (t)X (t􀀀)+A(t)]dt + [b (t)X (t􀀀 ∫ )+B(t)]dW (t) + R0 [g (t; z)X (t􀀀)+G(t; z)] ¯N (dt;dz) with a set of stochastic continuous functions fa;b ;g ;A;B;Gg and assuming that the compensated Poisson process ¯N (t; z) is independent of the Wiener process W(t). Derived from the Ito-Hermite formulas for the Ito-Hermite process and for the Ito-Levy process class we presented the results for the differential and multiple stochastic integration for the Ito- Hermite process. We also provided a separation method to solve jump-diffusion linear differential equations.

This paper is devoted to studying the p-th moment exponential stability for a class of stochastic differential equation (SDE) driven by G-Brownian motion under non-Lipschitz condition. The delays considered in this paper are time-varying delays τ i t 1 ≤ i ≤ 3 . Since the coefficients are non-Lipschitz, the normal enlargement on the coefficients is not available and the Gronwall inequality is not suitable in this case. By Bihari inequality and Itô integral formula, it is pointed out that there exists a constant τ ∗ such that the p-th moment exponential stability holds if the time-varying delays are smaller than τ ∗ .

Finding the optimal iteration of Gaussian quadrature rule is one of the important problems in the computational methods. In this study, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library to find the optimal iteration and optimal approximation of the Gauss-Legendre integration rule (G-LIR). A theorem is proved to show the validation of the presented method based on the concept of the common significant digits. Applying this method, an improper integral in the solution of the model of the osmosis system is evaluated and the optimal results are obtained. Moreover, the accuracy of method is demonstrated by evaluating other definite integrals. The results of examples illustrate the importance of using the stochastic arithmetic in discrete case in comparison with the common computer arithmetic.

This article presents a Lyapunov precise integral-based analysis method for seismic structures with added viscous fluid dampers. This study uses the full stress algorithm as the optimization method, considering the mean square of interstory drifts as the optimization objective, the position of the damper as the optimization object, and the random vibration analysis method as the calculation method to optimize seismic frame structures with viscous dampers. A precise integral solution is derived for the Lyapunov equation based on the general expression of the Lyapunov differential equation for the damping system under the excitation of a nonstationary stochastic process using two types of modulation functions: g(t)=1 and g(t)=t. Finally, the optimal damping arrangement is achieved using this method with a six-layer non-eccentric planar frame. In addition, the optimization results of this study are verified with those in the literature using time-history analysis, which verifies the feasibility and effectiveness of the proposed method. This study provides a method for the optimal configuration of dampers for seismic response of structures, which is beneficial for engineering applications and the protection of seismic structures.

SummaryWe introduce and study a new metric on denned bywhere is the space of probability measures on ℝk and where g: ℝk→ is a probability density satisfying certain mild conditions. The metric dg, relatively easy to compute, is shown to have useful and interesting properties not enjoyed by some other metrics on . In particular, letting pn denote the nth empirical measure for P, it is shown that under appropriate conditions satisfies a compact law of the iterated logarithm, converges in probability to the supremum of a Gaussian process, and has a useful stochastic integral representation.

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space ℱ ext (ℋ) over ℋ= L2(ℝd, dσ) and show how to include the usual Fock space ℱ(ℋ) in it as a subspace. We introduce in ℱ ext (ℋ) operators a(ξ)=∫ℝddxξ(x)a(x), ξ∈ S, with [Formula: see text], where [Formula: see text] and ∂x are the creation and annihilation operators at x. We show that (a(ξ))ξ∈S is a family of commuting self-adjoint operators in ℱ ext (ℋ) and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary I between ℱ ext (ℋ) and the L2-space L2(S', dμ G ), where μ G is the measure of Gamma white noise with intensity σ. The image of a(ξ) under I is the operator of multiplication by <·,ξ>, so that a(ξ)'s are Gamma field operators. The Fock structure of the Gamma space determined by I coincides with that discovered in Ref. 22. We note that I extends in a natural way the multiple stochastic integral (chaos) decomposition of the "chaotic" subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.

Stochastic medium (SM) theory is a practical method in ground settlement prediction, while its nonintegrable double integral form makes the solution process complicated. A simplified analytical solution based on the SM theory is developed to predict the ground movement in tunneling excavation. With the simplified solution, the ground movement for single tunnel and twin tunnels could be predicted based on the gap parameter G and influence angle β. A feasible approach is developed to estimate these two parameters using the maximum ground settlement Smax and tunnel design parameters, including tunnel depth H and diameter R. The proposed approach can be used to predict the ground movement curve for both circular and noncircular cross section tunnels. To validate its accuracy, the results predicted by the simplified procedure are compared with those obtained by the SM theory and measured in situ. The comparisons show that the current results agree well with those obtained by the SM theory and measured in situ. The comparison of five tunnels in literature illustrates that the simplified method can provide a more reasonable prediction for the ground movement induced by tunneling.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λn, n = 1,2,…, as well as an integral over the continuous spectrum above μ2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of orderν&gt; −1 with constant driftμ≠ 0. Whenν&gt; -½ andμ&lt; 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 &lt;ν&lt; -½ andμ&lt; 0, there is only one eigenvalueλ0= 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

We discuss a class of mathematical models of biological systems at microscopic level - i.e. at the level of interacting individuals of a population. The class leads to partially integral stochastic semigroups- [5]. We state general conditions guaranteeing the asymptotic stability. In particular under some rather restrictive assumptions we observe that any, even non-factorized, initial probability density tends in the evolution to a factorized equilibrium probability density - [4]. We discuss possible applications of the general theory such as redistribution of individuals - [2], thermal denaturation of DNA [1], and tendon healing process - [3]. [1] M. Debowski, M. Lachowicz, and Z. Szymanska, Microscopic description of DNA thermal denaturation, to appear. [2] M. Dolfin, M. Lachowicz, and A. Schadschneider, A microscopic model of redistribution of individuals inside an 'elevator', In Modern Problems in Applied Analysis, P. Drygas and S. Rogosin (Eds.), Bikhauser, Basel (2018), 77--86; DOI: 10.1007/978--3--319--72640-3.[3] G. Dudziuk, M. Lachowicz, H. Leszczynski, and Z. Szymanska, A simple model of collagen remodeling, to appear.[4] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on--line, DOI: 10.1002/mma.4680[5] K. Pichor and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Analysis Appl. 249, 2000, 668--685, DOI: 10.1006/jmaa.2000.6968

There is considerable evidence from around the world to support the idea that access to formal financial services is a key factor towards achieving poverty alleviation. The government of Indonesia has placed high importance on the issue of improved access to financial services and one feels that it would be appropriate to begin the process of analyzing effective financial inclusion initiatives with the existing Bank Perkreditan Rakyat (BPR) system. BPRs have long been an integral part of Indonesia’s financial, economic, and social development. This research is focused on BPRs in West Java. In this research, the writer evaluates BPR performance within twenty-five districts in West Java, by measuring the technical efficiency levels of the BPRs through employing the Stochastic Frontier Approach (SFA). The district that has the highest BPR average inefficiency score is Bandung city and the district with the lowest average BPR inefficiency score is the district of Ciamis. Increases in bank concentration (indicated by the Hirschman-Herfindahl Index) and income per capita are shown to decrease BPR inefficiency levels. Increases in the percentage of the population under the poverty line, the percentage of the labor force with a high school education, the percentage of road length per area, and the amount of bank offices per district, increase the inefficiency levels of BPRs. Overall, it is found that BPRs cannot operate efficiently in areas which are too underdeveloped; neither can they operate efficiently in areas which are too well developed.AbstrakTerdapat bukti substantif pengalaman berbagai negara di dunia bahwa akses pada jasa finansial formal adalah faktor kunci dalam upaya pengentasan kemiskinan. Pemerintah Republik Indonesia telah menempatkan upaya peningkatan mutu dan akses jasa finansial bagi masyarakat Indonesia, sebagai salah satu kebijakan utama dalam pengentasan kemiskinan. Tulisan ini bertujuan untuk membahas dan menganalisa efektivitas badan-badan penyedia jasa finansial di Indonesia, sebagai starting-point dengan memulai analisa efektivitas jasa finansial yang diberikan oleh Bank Perkreditan Rakyat (BPR). Penelitian ini difokuskan kepada BPR di Jawa Barat. Penulis mengevaluasi performa BPR pada 25 kabupaten/kota di Jawa Barat, dengan mengukur tingkat efisiensi teknis melalui metode Stochastic Frontier Approach (SFA). Kota tempat BPR yang memiliki rata-rata in-efisiensi tertinggi adalah Kota Bandung dan Kabupaten dengan in-efisiensi rata-rata BPR-nya yang terendah adalah Kabupaten Ciamis. Peningkatan dalam konsentrasi perbankan di sebuah kabupaten/kota (terindikasi Hirschman-Herfindahl Index) dan peningkatan pendapatan per kapita pada kabupaten/kota tersebut terbukti menurunkan tingkat inefisiensi BPR. Sebagai kesimpulan utama, ditemukan bahwa BPR tidak dapat beroperasi secara efektif pada area yang sangat underdeveloped dan juga tidak dapat beroperasi efektif pada area yang sangat developed. Kata kunci: Bank Perkreditan Rakyat (BPR); Jawa Barat; Efisiensi Teknis; Stochastic Frontier Approach (SFA)JEL classifications: D; L1; L2; G; G2

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

Abstract Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α-Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach (2016) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8, 237–266 (2005)) and, at the same time, reveals striking parallels.

AbstractThe symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor $$g_{\mu \nu }$$ g μ ν on an additional time variable, named intrinsic time at variance with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of $$g_{\mu \nu }$$ g μ ν quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative $${\dot{g}}_{\mu \nu }$$ g ˙ μ ν of the metric field with respect to intrinsic time, corresponding to the conjugated momentum $$\pi _{\mu \nu }$$ π μ ν . Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional $${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$ A [ g μ ν , π μ ν ] = K [ g μ ν , π μ ν ] - S [ g μ ν ] , playing formally the role of a Hamilton function, where $$S[g_{\mu \nu }]$$ S [ g μ ν ] is the standard Einstein–Hilbert action while $${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$ K [ g μ ν , π μ ν ] is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of $$g_{\mu \nu }$$ g μ ν analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action $${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$ A [ g μ ν , π μ ν ] and not of energy. Since the Einstein–Hilbert action $$S[g_{\mu \nu }]$$ S [ g μ ν ] plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of $$g_{\mu \nu }$$ g μ ν . Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field $$\pi _{\mu \nu }$$ π μ ν gives rise to a cosmological constant term in the path-integral approach.

In this article, a numerical-analytical model of the reflection coefficient of an electromagnetic wave from a rough soil surface is proposed. Based on the created model of reflection coefficient, in the frequency range from 520 MHz to 1.26 GHz, a broadband method for the root-mean-square deviation (RMS) of heights and volumetric moisture of rough soil surface is developed. When the model of reflection coefficient creating, the field of reflected wave from the rough soil surface was presented as the sum of secondary fields from an infinite set of elementary horizontal scattering plates (with a given average size). The field from each elementary scattering plate is represented as an average field from the result of stochastic interference of an infinite set of coherent elementary sources of secondary waves. The vertical position of the elementary sources is determined by the stochastic height of the soil surface within elementary plate. It is shown that the created model describes the total value of reflection coefficient (coherent and diffuse components) with the determination coefficient of R2=0.981 and root-mean square error (RMSE) no more than 0.35 dB, relative to the reflection coefficient, calculated by the finite difference method and the advanced integral equations method. The possibility of RMS heights and volumetric moisture of rough soil surface retrieval, relative to the true values, was shown no worse than R2=0.909 (RMSE=0.4 cm) and R2=0.975 (RMSE=2%), respectively. This analysis was carried out on the example of 16 soil samples, the complex permittivity of which was measured for a wide range of dry bulk densities from 0.7 g/cm3 to 1.8 g/cm3, moisture content, texture with variation clay content from 0% to 76% and organic matter content from 0.6% to 6.9%. The results have a wide application value both for single-frequency and multi-frequency radar and radiometric methods of soil moisture measuring.

The objective of this paper is to investigate natural transform of Meijer’s G-Function in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Abstract In 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.

In this paper, we analyze an underlay two-way decode-and-forward scheme in which secondary relays use successive interference cancellation (SIC) technology to decode data of two secondary sources sequentially, and then generate a coded signal by superposition coding (SC) technology, denoted as SIC-SC protocol. The SIC-SC protocol is designed to operate in two time slots under effects from an interference constraint of a primary receiver and residual interference of imperfect SIC processes. Transmit powers provided to carry the data are allocated dynamically according to channel powers of interference and transmission, and a secondary relay is selected from considering strongest channel gain subject to increase in decoding capacity of the first data and decrease in collection time of channel state information. Closed-form outage probability expressions are derived from mathematical manipulations and verified by performing Monte Carlo simulations. An identical scheme of underlay two-way decodeand-forward relaying with random relay selection and fixed power allocations is considered to compare with the proposed SIC-SC protocol, denoted as RRS protocol. Simulation and analysis results show that the non-identical outage performances of the secondary sources in the proposed SIC-SC protocol are improved by increasing the number of the secondary relays and the interference constraint as well as decreasing the residual interference powers. Secondly, the performance of the nearer secondary source is worse than that of the farther secondary source. In addition, the proposed SIC-SC protocol outperforms the RRS comparison protocol, and effect of power allocations through channel powers is discovered. Finally, derived theory values are precise to simulation results. Keywords: Successive interference cancellation, superposition coding, power allocation, underlay cognitive radio, non-orthogonal multiple access, outage probability. References [1] Popovski, H. Yomo, Physical Network Coding in Two-Way Wireless Relay Channels, presented at 2007 IEEE International Conference on Communications (ICC), Glasgow, 2007, pp. 707-712. https://doi.org/10.1109/ICC.2007.121. [2] Cao, X. Ji, J. Wang, S. Zhang, Y. Ji, J. Wang, Security-Reliability Tradeoff Analysis for Underlay Cognitive Two-Way Relay Networks, IEEE Transactions on Wireless Communications 18(12) (2019) 6030-6042. https://doi.org/10.1109/ TWC.2019.2941944. [3] Mitola, G.Q. Maguire, Cognitive radio: making software radios more personal, IEEE Personal Communications 6(4) (1999) 13-18. https://doi. org /10.1109/98.788210. [4] M.C. Chu, H. Zepernick, Performance Optimization for Hybrid Two-Way Cognitive Cooperative Radio Networks With Imperfect Spectrum Sensing, IEEE Access 6 (2018) 70582-70596. https://doi.org/10.1109/ICC.2007.121. [5] Ho-Van, T. Do-Dac, Security Analysis for Underlay Cognitive Network with Energy-Scavenging Capable Relay over Nakagami-m Fading Channels, Wireless Communications and Mobile Computing 2019 1-16. https://doi.org/ 10.1155/2019/5080952. [6] Zhang, Z. Zhang, J. Xing, R. Yu, P. Zhang, W.Wang, Exact Outage Analysis in Cognitive Two-WayRelay Networks With Opportunistic Relay SelectionUnder Primary User’s Interference, IEEE Transactionson Vehicular Technology 64(6) (2015) 2502-2511. https://doi.org/10.1109/2014.2346615. [7] T. Duy, H.Y. Kong, Exact outage probability of cognitive two-way relaying scheme with opportunistic relay selection under interference constraint, IET Communications 6(16) (2012), 2750-2759. https://doi.org/ 10.1049/iet-com. 2012.0235. [8] V. Toan, V.N.Q. Bao, Opportunistic relaying for cognitive two-way network with multiple primary receivers over Nakagami-m fading, presented at 2016 International Conference on Advanced Technologies for Communications (ATC), Hanoi city, 2016, pp.141-146. https://doi.org/1109/ATC.2016.7764762. [9] V. Toan, V.N.Q. Bao, H. Nguyen-Le, Cognitive two-way relay systems with multiple primary receivers: exact and asymptotic outage formulation, IET Communications 11(16) (2017) 2490-2497. https://doi.org/10.1049/iet-com.2017. 0400. [10] V.Toan, V.N.Q. Bao, K.N. Le,Performance analysis of cognitive underlay two-wayrelay networks with interference and imperfect channelstate information, EURASIP Journal on WirelessCommunications and Networking 2018 53 (2018).https://doi.org/10.1186/s13638-018-1063-z. [11] Solanki, P.K. Sharma, P.K. Upadhyay,Adaptive Link Utilization in Two-Way SpectrumSharing Relay Systems Under Average Interference Constraints, IEEE Systems Journal 12(4) (2018) 3461-3472. https://doi.org/10.1109/ JSYST.2017.2713887. [12] Yue, Y. Liu, S. Kang, A. Nallanathan, Y. Chen, Modeling and Analysis of Two-WayRelay Non-Orthogonal Multiple Access Systems, IEEETransactions on Communications 66(9) (2018) 3784-3796. https://doi.org/10.1109/TCOMM. 2018.2816063. [13] Zou, B. He, H. Jafarkhani, An Analysis of TwoUser Uplink Asynchronous Non-orthogonal MultipleAccess Systems, IEEE Transactions on WirelessCommunications 18(2) (2019) 1404-1418. https://doi.org/10.1109/TWC.2019.2892486. [14] Yang, Z. Ding, P. Fan, N. Al-Dhahir, TheImpact of Power Allocation on Cooperative Nonorthogonal Multiple Access Networks With SWIPT,IEEE Transactions on Wireless Communications 16(7) (2017) 4332-4343. https://doi.org/10.1109/TWC.2017.2697380. [15] N. Son, T.T. Duy, K. Ho-Van, SIC-Coding Schemes for Underlay Two-Way Relaying Cognitive Networks, Wireless Communications and Mobile Computing 2020, pp.1-24. https://doi.org/ 10.1155/2020/8860551. [16] F. Kader, M.B. Shahab, S.Y. Shin, ExploitingNon-Orthogonal Multiple Access in Cooperative RelaySharing, IEEE Communications Letters 21(5) (2017) 1159-1162. https://doi.org/1109/LCOMM.2017.2653777. [17] Yue, Y. Liu, S. Kang, A. Nallanathan, Z. Ding,Spatially Random Relay Selection for Full/Half-DuplexCooperative NOMA Networks, IEEE Transactions onCommunications 66(8) (2018) 3294-3308. https://doi.org/10.1109/TCOMM. 2018.2809740. [18] Liu, Z. Ding, M. Elkashlan, J. Yuan,Nonorthogonal Multiple Access in Large-Scale UnderlayCognitive Radio Networks, IEEE Transactions onVehicular Technology 65(12) (2016)10152-10157. https://doi.org/10.1109/ TVT.2016.2524694. [19] Song, W. Yang, Z. Xiang, N. Sha, H. Wang, Y.Yang, An Analysis on Secure Millimeter Wave NOMACommunications in Cognitive Radio Networks, IEEE Access 8 (2020), 78965-78978. https://doi.org/10.1109/ACCESS.2020.2989860. [20] Ding, T. Song, Y. Zou, X. Chen, L. Hanzo,Security-Reliability Tradeoff Analysis of Artificial NoiseAided Two-Way Opportunistic Relay Selection, IEEE Transactions on Vehicular Technology 66(5) (2017) 3930-3941. https://doi.org/10.1109/TVT.2016.2601112. [21] Zheng, M. Wen, F. Chen, J. Tang, F. Ji, SecureNOMA Based Full-Duplex Two-Way Relay Networkswith Artificial Noise against Eavesdropping, presented at 2018IEEE International Conference on Communications(ICC), Kansas City, 2018,pp.1-6. https://doi.org/ 10.1109/ICC.2018.8422946. [22] N. Son,H.Y. Kong, Exact Outage Analysisof Energy Harvesting Underlay Cooperative CognitiveNetworks, IEICE Transactions on Communications E98-B(4) (2015),pp.661-672. https://doi.org/10.1587/transcom.E98.B.661. [23] Tourki, K.A. Qaraqe, M. Alouini, OutageAnalysis for Underlay Cognitive Networks UsingIncremental Regenerative Relaying, IEEE Transactions on Vehicular Technology 62(2) (2013) 721-734. https://doi.org/10.1109/TVT. 2012.2222947. [24] Papoulis, S.U. Pillai, Probability, randomvariables and stochastic processes, 4th ed., McGrawHill, New York, 2002. [25] Pei, T. Zhifeng, L. Zinan, E. Erkip, S.Panwar, Cooperative wireless communications: a cross-layer approach, IEEE Wireless Communications 13(4) (2006) 84-92. https://doi.org/10.1109/2006.1678169. [26] Ghasemi, E.S. Sousa, Fundamental limitsof spectrum-sharing in fading environments, IEEETransactions on Wireless Communications 6(2) (2007) 649-658. https://doi.org/10.1109/TWC. 2007.05447. [27] M. Peha, Approaches to spectrum sharing, IEEECommunications Magazine 43(2) (2005) 10-12. https://doi.org/10.1109/MCOM.2005. 1391490. [28] Kim, S. Lim, H. Wang, D. Hong, Optimal PowerAllocation and Outage Analysis for Cognitive FullDuplex Relay Systems, IEEE Transactions on Wireless Communications 11(10) (2012) 3754-3765. https://doi.org/10.1109/TWC. 2012.083112.120127. [29] N. Son,T.T. Duy, Performance analysisof underlay cooperative cognitive full-duplexnetworks with energy-harvesting relay, ComputerCommunications 122 (2018) 9-19. https://doi.org/1016/j.comcom.2018.03.003. [30] V. Nguyen, T. Do, V.N.Q. Bao, D.B.d.Costa, B. An, On the Performance of MultihopCognitive Wireless Powered D2D Communications inWSNs, IEEE Transactions on Vehicular Technology 69(3) (2020) 2684-2699. https://doi.org/10.1109/TVT.2020.2963841. [31] Ruan, Y. Li, C. Wang, R. Zhang, H.Zhang, Energy Efficient Power Allocation for DelayConstrained Cognitive Satellite Terrestrial NetworksUnder Interference Constraints, IEEE Transactions on Wireless Communications 18(10) (2019) 4957-4969. https://doi.org/10.1109/TWC. 2019.2931321. [32] Gao, S. Zhang, Y. Su, M. Diao, M. Jo, Joint Multiple Relay Selection and Time Slot Allocation Algorithm for the EH-Abled Cognitive Multi-User Relay Networks, IEEE Access 7 (2019) 111993- 112007. https://doi.org/10.1109/2019.2932955. [33] Arezumand, H. Zamiri-Jafarian, E. Soleimani-Nasab, Exact and Asymptotic Analysis of Partial Relay Selection for Cognitive RF-FSO Systems With Non-Zero Boresight Pointing Errors, IEEE Access 7 (2019) 58611-58625. https://doi.org/1109/ACCESS.2019.2914480. [34] N. Son, H.Y. Kong, Energy-Harvesting Relay Selection Schemes for Decode-and-Forward Dual-Hop Networks, IEICE TRANSACTIONS on Communications E98-B(12) (2015) 2485-2495. https://doi.org/10.1587/transcom.E98.B.2485. [35] N. Nguyen, T.H. Quang Minh, P.T. Tran, M. Voznak, T.T. Duy, T.-L. Nguyen, P.T. Tin, Performance enhancement for energy harvesting based two-way relay protocols in wireless ad-hoc networks with partial and full relay selection methods, Ad Hoc Networks 84 (2019) 178-187. https://doi.org/10.1016/j.adhoc.2018.10.005. [36] Pan, Z. Li, Z. Wang, F. Zhang, Joint Relay Selection and Power Allocation for the Physical Layer Security of Two-Way Cooperative Relaying Networks, Wireless Communications and Mobile Computing, 2019, pp. 1-7. https://doi.org/10.1155/2019/1839256. [37] A. Nasir, Z. Xiangyun, S. Durrani, R.A. Kennedy, Relaying Protocols for Wireless Energy Harvesting and Information Processing, IEEE Transactions on Wireless Communications 12(7) (2013) 3622-3636. https://doi.org/10.1109/TWC. 2013.062413.122042. [38] I. Gradshteyn, I.M. Ryzhik, A.Jeffrey, D. Zwillinger, Table of integral, series and products, 7th ed., Elsevier, Amsterdam, 2007. [39] Haiyan, L. Zan, S. Jiangbo, G. Lei, Underlay cognitive relay networks with imperfect channel state information and multiple primary receivers, IET Communications 9(4) (2015) 460-467. https://doi.org/10.1049/iet-com.2014.0429. [40] Zhong, Z. Zhang, Opportunistic Two-Way Full-Duplex Relay Selection in Underlay Cognitive Networks, IEEE Systems Journal 12(1) (2018) 725-734.https://doi.org/10.1109/JSYST. 2016.2514601.