In this paper, we shall ﬁrstly illustrate why we should consider **integral** of a **stochastic** process with **respect** to a **set**-**valued** **square** **integrable** martingale. Secondly, we shall prove the representation theorem of **set**-**valued** **square** **integrable** martingale. Thirdly, we shall give the deﬁnition of **stochastic** **integral** of a **stochastic** process with **respect** to a **set**-**valued** **square** **integrable** martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the **stochastic** **integral** is a **set**-**valued** sub-martingale.

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In this paper, we formulate an equivalent definition of the Itˆ o-Henstock **integral** of an operator-**valued** **stochastic** process with **respect** to a Hilbert space-**valued** Q-Wiener process. To attain this objective, we use the concept of the double Lusin condition and AC 2 [0, T ]-property, a version of absolute continuity. A worthwhile direction for further investigation is to use Henstock-Kurzweil approach to define the **stochastic** **integral** with **respect** to a cylindrical Wiener process.

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2000 Mathematics Subject Classification. $M$ -type 2 Banach space, set-valued stochastic $proc-$ , Set valued stochastic integral,.. Poisson random measure, Compensated Poisson random me[r]

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In this paper, different from the definition in [14], based on the Definition 3.1 in [12], we will study the Lebesgue-Stieltjes **integral** of **set**-**valued** **stochastic** pro- cesses with **respect** to single **valued** finite variation pro- cess. We shall prove the measurability of **integral**, namely, it is a **set**-**valued** random, which is similar to the classical **stochastic** **integral**.

In infinite dimensional spaces, the Itˆ o **integral** of an operator-**valued** **stochastic** process, adapted to a normal filtration, is obtained by extending an isometry from the space of elementary processes to the space of continuous **square**-**integrable** **martingales**. In this case, the value of the integrand is an operator and the integrator is a Q-Wiener process, a Hilbert space-**valued** Wiener process which is dependent on a symmetric nonnegative definite trace-class operator Q.

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much worse than for the other treatments. This procedure can deal with the case where the two interested outcome have different units (e.g., the amount of reduction in blood pressure on the one hand and in blood fat on the other). Note, however, that giving a precise meaning to “not too much worse” requires a user specified threshold. The main contribution of this work is twofold. First, we for the first time consider **set** **valued** DTRs when the interested outcome is a scaler without any user-specified threshold. To accommodate the **set**-**valued** DTRs we give a new definition of the effect of a non-final stage treatment, which usually is a random quantity instead of a fixed quantity. Second, we propose a new scientific goal of DTRs: instead of a DTR attempting to pin down the true best treatment, it should screen out significantly worse treatments for a given patient.

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Recent advances in machine learning, particularly deep learning models and training algorithms, have resulted in significant breakthroughs in a variety of AI areas, including computer vision, natural language processing, and speech recognition. Most of these applications have been formu- lated as classification problems: a label is predicted for a given input. The output label could be the category of an image, the word uttered in an audio signal, or the topic of a news paragraph. For sequence generation problems, an or- dered list of tokens is generated sequentially, with the output of each token being essentially a label prediction. In this pa- per, we pursue the capability to predict sets, the size of which may vary, and for which the order of the elements is irrele- vant. We call this problem **set** prediction. The challenge lies in the fact that the output space, or the universe of **set** ele- ments, may be enormously large or even infinite, especially for sets of sequences. Thus, treating the general problem as multi-label classification is inefficient or effectively impos- sible. Examples of **set** prediction problems include learning Copyright c 2019, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

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The vector criterion and **set** criterion are two deﬁning approaches of solutions for the **set**-**valued** optimization problems. In this paper, the optimality conditions of both criteria of solutions are established for the **set**-**valued** optimization problems. By using Studniarski derivatives, the necessary and suﬃcient optimality conditions are derived in the sense of vector and **set** optimization.

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Reference [1] gives Riesz decomposition of **set**-**valued** supermartingale in real space and promotes the results to reflexive Banach spaces (reference [2]). Reference [2] gives the counter-example that not all of the **set**-**valued** martingale has Riesz decomposition in a two-dimensional plane case. The fundamental reason is the defects of algebraic operation on hyperspace. Therefore, the research can pursue the unstrict sense of Riesz decomposition instead of studying various sense of Riesz decomposition. Reference [2] shows the other Riesz decomposition of **set**-**valued** supermartingale in real space. Reference [3] gives Riesz decomposition of **set**-**valued** supermartingale in the general Banach space under the X * separable condition. References [4] and [5] research Riesz decomposi- tion of **set**-**valued** submartingale in the general Banach space. Reference [6] studies Riesz decomposition in weak **set**-**valued** Amart. Reference [7]-[9] gives every sense of Riesz decomposition of **set**-**valued** Pramart in the general Banach space under the X * separable condition. Reference [10] studies the problems of Riesz decomposi- tion of **set**-**valued** Pramart. All of the above studies have given the necessary and sufficient conditions for Riesz decomposition. The research of every sense of Riesz decomposition of **set**-**valued** Superpramart is still rare.

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It has been reported that real-**valued** neural networks (RVNNs) are useful in various scientific fields, for example, optimization, image and signal processing, as well as associative memory [ 10 , 16 , 17 ]. However, RVNN models perform poorly in tackling the XOR problem and in 2D affine transformations [ 18 , 19 ]. In view of this, complex properties have been incorporated into RVNNs to formulate complex-**valued** neural networks (CVNNs), which can effectively solve the XOR problem and 2D affine transformations [ 18 , 19 ]. As a result, CVNN-related models have received significant research attention in both mathematical and practical analyses [ 20 – 24 ]. Nevertheless, CVNN models are inefficient in handling higher dimension transformations including color night vision, color image compression, and 3D and 4D problems [ 25 – 27 ]. Meanwhile, several engineering problems involve quaternion-**valued** signals and quaternion functions, such as 3D wind forecasting, polarized signal classification, color night vision, as well as color night vision [ 26 – 28 ]. Undoubtedly, quaternion-based networks present good mathematical models to undertake these applications due to the quaternion features. In view of this, quaternion-**valued** neural networks (QVNNs) have been developed by implementing quaternion algebra into CVNNs, in order to generalize RVNN and CVNN models with quaternion-**valued** activation functions, connection weights, as well as signals states [ 11 , 29 – 32 ]. Therefore, the investigation of the dynamics of QVNN models is essential and important. Recently, many computational approaches for various QVNN models and their learning algorithms have been proposed. Among the studies include global µ stability, global asymptotic stability as well as global synchronization [ 29 – 31 ]. Other studies of QVNN models are also available, for example, exponential input-to-state stability (exp-ISS) and global Mittag-Leffler stability and synchronization [ 32 , 33 ]. Similarly, some other stability conditions have been defined for QVNN models [ 12 , 34 , 35 ].

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We have been carrying out research on a new framework which involves an approach that clearly established the relationship between ‘Feynmannian path **integral**’ with the Feynman **integral** and real/ordinary **integral** without the use of limiting procedures in a generalized space for a general class of potentials (refer Shaharir 1986, Shaharir 1995, Shaharir & Zainal 1995, 1996a & b, Zainal 2001). We too are looking back into this framework (for consolidating purposes) in the context of connecting it with various basic and current concepts in (physically significant) completely **integrable** systems (refer Zainal 1998, 2004a, b, c). Accordingly we are listing down four mainstream issues in this field that we had successfully carried out research, currently pursuing and planning to explore further. For the purpose of our proposed SAGA grant application, we would be largely concentrating our efforts to unravel the fourth issue as stated below, which is being researched intensively since the end of last century (refer Dubrovin, 1996).

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contribution to the **integral** in (l.l) due to 0 values distant from 0Q can be neglected. Thus we require (e.g.) consistency of the strict MLE(s), and a simple way of ensuring this is to assume (in addition to Wald's conditions) that L^(0) = 0 has a unique solution with probability tending to unity. From the point of view of applications, this is not usually a restric tion, for often the uniqueness is clear.

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Existence and Uniqueness of Solutions 1.1 Introduction 1.2 Basic Setting and Assumptions 1.3 Existence and Uniqueness of Solutions 1.4 Generalizations 1.5 Applications to Doleans-Dade's [r]

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pseudodistance which is an extension of the b-metric. Next, inspired by the ideas of Nadler (Pac. J. Math. 30:475-488, 1969) and Abkar and Gabeleh (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2):319-325, 2013), we deﬁne a new **set**-**valued** non-self-mapping contraction of Nadler type with **respect** to this b-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for T : A → 2 B . A best proximity point theorem furnishes suﬃcient conditions that

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This paper studies ﬁrst a result of existence and uniqueness of the solution to a backward **stochastic** diﬀerential equation driven by an inﬁnite dimensional martingale. Then, we apply this result to ﬁnd a unique solution to a backward **stochastic** partial diﬀerential equation in inﬁnite dimensions. The ﬁltration considered is an arbitrary right- continuous ﬁltration, not necessarily the natural ﬁltration of a Wiener process. This, in particular, allows us to study more applications, for example the maximum principle for a controlled **stochastic** evolution system. Some examples are discussed in the paper as well.

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Indeed, quite recently, risk measures have been studied and extended to a more general setting where they can be **set**-**valued** ([23], [20]): one among the different financial motivations for studying **set**-**valued** risk measures is given by the existence of portfolios of financial positions in different currencies that can not be aggregated for reasons such as liquidity constraints and/or transaction costs ([23], [20]). In this case, in fact, it seems more reasonable to consider risk measures that associate to any financial portfolio in different currencies a **set** of hedging deterministic positions. Dual representations for **set**-**valued** risk measures can be found, among many others, in [19], [20] while extensions to the dynamic framework can be found in [12], [13], [4]. Recently, many well known risk measures have been extended to the **set**- **valued** case (see, for example [1], [15], [25]) and, furthermore, **set**-**valued** risk measures have also been applied to the study of systemic risk ([2],[14]).

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The INAR(1) model as discussed in Chapters 4 and 5 can be seen as an integer- **valued** version of the continuous AR(1) model, which is quite easily extended to a higher order AR(p) model, cf. (2.18). It is therefore not surprising that the modeling of higher order autoregressive structures has been considered in the literature. Indeed, the first contributions considering INAR(p) processes (with p > 1) were published shortly after the introduction of the INAR(1) model in McKenzie (1985) and Al-Osh and Alzaid (1987). However, the two most prominent attempts, given by Alzaid and Al-Osh (1990) and Du and Li (1991), differ substantially so that there exists no canonical extension of the INAR(1) process. In terms of popularity, there does not seem to be a large difference between the two formulations when comparing the number of times these articles were cited. As of this writing (April 2015), the web page scholar.google.com reports 161 citations for the article Alzaid and Al-Osh (1990) and 166 for Du and Li (1991).

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The purpose of this paper is to study a new class of fuzzy nonlinear **set**-**valued** variational inclusions in real Banach spaces. By using the fuzzy resolvent operator techniques for m-accretive mappings, we establish the equivalence between fuzzy nonlinear **set**-**valued** variational inclusions and fuzzy resolvent operator equation problem. Applying this equivalence and Nadler’s theorem, we suggest some iterative algorithms for solving fuzzy nonlinear **set**-**valued** variational inclusions in real Banach spaces. By using the inequality of Petryshyn, the existence of solutions for these kinds of fuzzy nonlinear **set**-**valued** variational inclusions without compactness is proved and convergence criteria of iterative sequences generated by the algorithm are also discussed.

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F A f Such a property is not shared by vector **valued** **set** functions. We introduce a suitable definition of the **integral** that will extend the above property to the vector **valued** case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive **set** functions with values in an infinite dimensional normed space.

formal expression is a some kind of the generalization of the notion of lo- cal time for the measure-**valued** processes. There are two important reasons which lead to the existence of the local time. First, the trajectories of the single particles, from which our process is composed, can have the usual local time at the point u. Second, our process consists of the smooth measures. We will consider the special type of the measure-**valued** processes, which is organized from the mass of the interacting particles. But in the case d ≥ 2, one cannot expect the existence of the local time for the trajectory of the single particle. Also we will use the singular initial mass distribution.

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