Statistical Methods in Mathematical Finance and Economics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 3452

Special Issue Editors


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Guest Editor
Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China
Interests: nonparametric estimation in mathematical finance

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Guest Editor
1. School of Mathematics, Shandong University, 27, Shanda South Road, Jinan 250100, China
2. Research Center for Mathematics and Interdisciplinary Sciences, 72 Binhai Road, Qingdao, China
Interests: mathematical finance

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Guest Editor
School of Finance and Business, Shanghai Normal University, Shanghai 200234, China
Interests: statistical method in high-frequency data

Special Issue Information

Dear Colleagues,

In the 1990s, Professor Peng Shige and others introduced the theory of nonlinear backward stochastic differential equations and the theory of nonlinear mathematical expectations, which have since been extensively promoted in financial mathematics research. As part of financial high-tech, financial mathematics can achieve significant progress in theoretical analysis and greatly promote research in applied fields such as empirical studies in finance. To encourage empirical financial studies, it is necessary to study statistical methods in financial mathematics. The study of statistical methods on high-frequency data is an essential part of statistical methods in financial mathematics. In addition to studying statistical methods of high-frequency data, this Special Issue will also explore the statistical methods under model uncertainty. We believe this Special Issue will significantly promote the study of statistical methods in financial mathematics.

Prof. Dr. Hanchao Wang
Prof. Dr. Zengjing Chen
Dr. Yuping Song
Guest Editors

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Keywords

  • central limit theorems
  • law of large numbers
  • sublinear expectations
  • two-armed bandit problems
  • large sample theorems of nonparametric statistics
  • sequential design
  • optimal strategy
  • high-frequency data

Published Papers (4 papers)

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Research

21 pages, 336 KiB  
Article
The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences under Sublinear Expectation
by Shuxia Guo and Zhe Meng
Mathematics 2023, 11(23), 4734; https://doi.org/10.3390/math11234734 - 22 Nov 2023
Viewed by 532
Abstract
In this paper we study the Marcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences under sublinear expectation. Specifically, we establish complete convergence in the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively dependent and identically distributed random variables under [...] Read more.
In this paper we study the Marcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences under sublinear expectation. Specifically, we establish complete convergence in the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively dependent and identically distributed random variables under certain moment conditions. We also give results for sequences of independent and identically distributed random variables. The moment conditions in this paper are based on a class of slowly varying functions that satisfy some convergence properties. Moreover, some special examples and comparisons to existing results are also given. Full article
(This article belongs to the Special Issue Statistical Methods in Mathematical Finance and Economics)
19 pages, 339 KiB  
Article
Optimal Investment–Consumption–Insurance Problem of a Family with Stochastic Income under the Exponential O-U Model
by Yang Wang, Jianwei Lin, Dandan Chen and Jizhou Zhang
Mathematics 2023, 11(19), 4148; https://doi.org/10.3390/math11194148 - 01 Oct 2023
Viewed by 718
Abstract
A household consumption and optimal portfolio problem pertinent to life insurance (LI) in a continuous time setting is examined. The family receives a random income before the parents’ retirement date. The price of the risky asset is driven by the exponential Ornstein–Uhlenbeck (O-U) [...] Read more.
A household consumption and optimal portfolio problem pertinent to life insurance (LI) in a continuous time setting is examined. The family receives a random income before the parents’ retirement date. The price of the risky asset is driven by the exponential Ornstein–Uhlenbeck (O-U) process, which can better reflect the state of the financial market. If the parents pass away prior to their retirement time, the children do not have any work income and LI can be purchased to hedge the loss of wealth due to the parents’ accidental death. Meanwhile, utility functions (UFs) of the parents and children are individually taken into account in relation to the uncertain lifetime. The purpose of the family is to appropriately maximize the weighted average of the corresponding utilities of the parents and children. The optimal strategies of the problem are achieved using a dynamic programming approach to solve the associated Hamilton–Jacobi–Bellman (HJB) equation by employing the convex dual theory and Legendre transform (LT). Finally, we aim to examine how variations in the weight of the parents’ UF and the coefficient of risk aversion affect the optimal policies. Full article
(This article belongs to the Special Issue Statistical Methods in Mathematical Finance and Economics)
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11 pages, 287 KiB  
Article
Exponential Inequality of Marked Point Processes
by Chen Li and Yuping Song
Mathematics 2023, 11(4), 881; https://doi.org/10.3390/math11040881 - 09 Feb 2023
Viewed by 671
Abstract
This paper presents the uniform concentration inequality for the stochastic integral of marked point processes. We developed a new chaining method to obtain the results. Our main result is presented under an entropy condition for partitioning the index set of the integrands. Our [...] Read more.
This paper presents the uniform concentration inequality for the stochastic integral of marked point processes. We developed a new chaining method to obtain the results. Our main result is presented under an entropy condition for partitioning the index set of the integrands. Our result is an improvement of the work of van de Geer on exponential inequalities for martingales in 1995. As applications of the main result, we also obtained the uniform concentration inequality of functional indexed empirical processes and the Kakutani–Hellinger distance of the maximum likelihood estimator. Full article
(This article belongs to the Special Issue Statistical Methods in Mathematical Finance and Economics)
19 pages, 332 KiB  
Article
On the Conjecture of Berry Regarding a Bernoulli Two-Armed Bandit
by Jichen Zhang and Panyu Wu
Mathematics 2023, 11(3), 733; https://doi.org/10.3390/math11030733 - 01 Feb 2023
Viewed by 967
Abstract
In this paper, we study an independent Bernoulli two-armed bandit with unknown parameters ρ and λ, where ρ and λ have a pair of priori distributions such that [...] Read more.
In this paper, we study an independent Bernoulli two-armed bandit with unknown parameters ρ and λ, where ρ and λ have a pair of priori distributions such that dR(ρ)=CRρr0(1ρ)r0dμ(ρ),dL(λ)=CLλl0(1λ)l0dμ(λ) and μ is an arbitrary positive measure on [0,1]. Berry proposed the conjecture that, given a pair of priori distributions (R,L) of parameters ρ and λ, the arm with R is the current optimal choice if r0+r0<l0+l0 and the expectation of ρ is not less than that of λ. We give an easily verifiable equivalent form of Berry’s conjecture and use it to prove that Berry’s conjecture holds when R and L are two-point distributions as well as when R and L are beta distributions and the number of trials Nr0r0+1. Full article
(This article belongs to the Special Issue Statistical Methods in Mathematical Finance and Economics)
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