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The Kurzweil integral in financial market modeling

Pavel Krejčí, Harbir Lamba, Giselle Antunes Monteiro, Dmitrii Rachinskii (2016)

Mathematica Bohemica

Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities...

The Kurzweil integral with exclusion of negligible sets

Pavel Krejčí (2003)

Mathematica Bohemica

We propose an extended version of the Kurzweil integral which contains both the Young and the Kurzweil integral as special cases. The construction is based on a reduction of the class of δ -fine partitions by excluding small sets.

The Kurzweil-Henstock theory of stochastic integration

Tin-Lam Toh, Tuan-Seng Chew (2012)

Czechoslovak Mathematical Journal

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive...

The L r Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

We present a method of integration along the lines of the Henstock-Kurzweil integral. All L r -derivatives are integrable in this method.

The M α and C -integrals

Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Deuk Ho Lee (2012)

Czechoslovak Mathematical Journal

In this paper, we define the M α -integral of real-valued functions defined on an interval [ a , b ] and investigate important properties of the M α -integral. In particular, we show that a function f : [ a , b ] R is M α -integrable on [ a , b ] if and only if there exists an A C G α function F such that F ' = f almost everywhere on [ a , b ] . It can be seen easily that every McShane integrable function on [ a , b ] is M α -integrable and every M α -integrable function on [ a , b ] is Henstock integrable. In addition, we show that the M α -integral is equivalent to the C -integral....

The McShane, PU and Henstock integrals of Banach valued functions

Luisa Di Piazza, Valeria Marraffa (2002)

Czechoslovak Mathematical Journal

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...

The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals

Márcia Federson (2002)

Czechoslovak Mathematical Journal

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, R d α ( t ) f ( t ) , where R is a compact interval of n , α and f are functions with values on L ( Z , W ) and Z respectively, and Z and W are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, R α ( t ) d f ( t ) , as well as to unbounded intervals R .

The s-Perron, sap-Perron and ap-McShane integrals

Joo Bong Kim, Deok Ho Lee, Woo Youl Lee, Chun-Gil Park, Jae Myung Park (2004)

Czechoslovak Mathematical Journal

In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.

The topology of the space of ℋ𝒦 integrable functions in n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm · X is defined. A · X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a · X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every control-convergent...

The Vitali convergence theorem for the vector-valued McShane integral

Richard Reynolds, Charles W. Swartz (2004)

Mathematica Bohemica

The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in n given by Kurzweil and...

The weak McShane integral

Mohammed Saadoune, Redouane Sayyad (2014)

Czechoslovak Mathematical Journal

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ -finite outer regular quasi Radon measure space ( S , Σ , 𝒯 , μ ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if...

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

We study the integrability of Banach space valued strongly measurable functions defined on [ 0 , 1 ] . In the case of functions f given by n = 1 x n χ E n , where x n are points of a Banach space and the sets E n are Lebesgue measurable and pairwise disjoint subsets of [ 0 , 1 ] , there are well known characterizations for Bochner and Pettis integrability of f . The function f is Bochner integrable if and only if the series n = 1 x n | E n | is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of f ....

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