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Real Line – An Incomplete Number System 实线-一个不完全的数字系统
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2021-07-26 DOI: 10.2139/ssrn.3893510
NM Ganguli
{"title":"Real Line – An Incomplete Number System","authors":"NM Ganguli","doi":"10.2139/ssrn.3893510","DOIUrl":"https://doi.org/10.2139/ssrn.3893510","url":null,"abstract":"The algebraic equation a3 + b3 = c3 where a, b & c are real numbers greater than zero, has infinite solutions. ‘R’ denotes the set of real numbers as well as the ‘Real Line’. The real numbers are treated as if those are points on the ‘Real Line’ and the points on the ‘Real Line’ as if those are real numbers. Hence a, b & c will have corresponding points on the Real Line and can be represented by three straight lines denoting three sides of a triangle, say ΔABC, on a two dimensional plane. Consequently the algebraic equation will yield the trigonometric equation Sin3A + Sin3B = Sin3C. Trigonometric properties of such a triangle i.e. DABC, imply that ÐC will be independent of (a & b) if c is constant; but solutions of the algebraic equation show that ÐC is dependent on (a & b) when c is constant, leading to a contradiction. Hence, all real numbers of the form ∛x cannot be considered as if those are points on the ‘Real Line’; and there are no points on the ‘Real Line’ corresponding to the irrational numbers of the form ∛x. This leads to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the ‘Real Line’.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114606456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Introduction of Extrapolated Block Adams Moulton Methods for Solving First-order Delay Differential Equations 求解一阶时滞微分方程的外推Block Adams Moulton方法
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2021-06-01 DOI: 10.22377/ajms.v5i1.307
C. Chibuisi, B. Osu, S. Ihedioha, C. Olunkwa, I. H. Onyekachukwu
{"title":"The Introduction of Extrapolated Block Adams Moulton Methods for Solving First-order Delay Differential Equations","authors":"C. Chibuisi, B. Osu, S. Ihedioha, C. Olunkwa, I. H. Onyekachukwu","doi":"10.22377/ajms.v5i1.307","DOIUrl":"https://doi.org/10.22377/ajms.v5i1.307","url":null,"abstract":"In this paper, the discrete schemes of extrapolated block Adams Moulton methods were obtained through the continuous formulation of the linear multistep collocation method by matrix inversion approach for the numerical solutions of first-order delay differential equations (DDEs) without the use of interpolation techniques in evaluating the delay term. The delay term was computed by a valid idea of sequence. The advantages, convergence, stability analysis, and central processing unit time at a constant step size bof the proposed method over other existing methods are pointed out.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127961244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Feasible Implied Correlation Matrices from Factor Structures 因子结构的可行隐含相关矩阵
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2021-05-11 DOI: 10.2139/ssrn.3876660
Wolfgang Schadner
{"title":"Feasible Implied Correlation Matrices from Factor Structures","authors":"Wolfgang Schadner","doi":"10.2139/ssrn.3876660","DOIUrl":"https://doi.org/10.2139/ssrn.3876660","url":null,"abstract":"Forward-looking correlations are of interest in different financial applications, including factor-based asset pricing, forecasting stock-price movements or pricing index options. With a focus on non-FX markets, this paper defines necessary conditions for option implied correlation matrices to be mathematically and economically feasible and argues that existing models are typically not capable of guaranteeing such. To overcome this difficulty, the problem is addressed from the underlying factor structure and introduces two approaches to solve it. Under the quantitative approach, the puzzle is reformulated into a nearest correlation matrix problem which can be used either as a stand-alone estimate or to re-establish positive-semi-definiteness of any other model’s estimate. From an economic approach, it is discussed how expected correlations between stocks and risk factors (like CAPM, Fama-French) can be translated into a feasible implied correlation matrix. Empirical experiments are carried out on monthly option data of the S&P 100 and S&P 500 index (1996-2020).","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129449166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adaptive Shrinkage Estimation of High Dimensional Moment Condition Model with Smooth Structural Changes 具有光滑结构变化的高维矩条件模型的自适应收缩估计
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2021-02-25 DOI: 10.2139/ssrn.3792704
Xingyi Chen, Yongmiao Hong, Haiqi Li
{"title":"Adaptive Shrinkage Estimation of High Dimensional Moment Condition Model with Smooth Structural Changes","authors":"Xingyi Chen, Yongmiao Hong, Haiqi Li","doi":"10.2139/ssrn.3792704","DOIUrl":"https://doi.org/10.2139/ssrn.3792704","url":null,"abstract":"Structural change is a long-standing problem in time series econometrics and macroeconomics, and financial time series are likely to be affected by structural instability due to changes in preferences, technologies, policies, etc. Most of the existing literature on moment condition models assumed that structural parameters were fixed over the entire sample period. To allow for time-varying structural parameters in the high dimensional moment condition models, this paper proposes a shrinkage local generalized method of moment (SLGMM) which simultaneously achieves parameter estimation and moment selection. We show that our method consistently selects the correct moment conditions and the SLGMM estimator possesses the oracle property, that is, it is as efficient as the time-varying GMM estimator based on all valid moment conditions. Moreover, we establish the consistency and asymptotic normality of the SLGMM estimator. A Monte Carlo simulation and an empirical application on asset pricing are conducted to show the merits of the newly proposed method.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117246924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dynamic Relocations in Car-Sharing Networks 汽车共享网络中的动态重新定位
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2021-01-27 DOI: 10.2139/ssrn.3774324
M. Hosseini, Joseph M. Milner, Gonzalo Romero
{"title":"Dynamic Relocations in Car-Sharing Networks","authors":"M. Hosseini, Joseph M. Milner, Gonzalo Romero","doi":"10.2139/ssrn.3774324","DOIUrl":"https://doi.org/10.2139/ssrn.3774324","url":null,"abstract":"We propose a novel dynamic car relocation policy for a car-sharing network with centralized control and uncertain, unbalanced demand. The policy is derived from a reformulation of the linear programming fluid model approximation of the dynamic problem. We project the full-dimensional fluid approximation onto the lower-dimensional space of relocation decisions only. This projection results in a characterization of the problem as n+1 linear programs, where n is the number of nodes in the network. The reformulation uncovers structural properties that are interpretable using absorbing Markov chain concepts and allows us to write the gradient with respect to the relocation decisions in closed form. Our policy exploits these gradients to make dynamic car relocation decisions. We provide extensive numerical results where our dynamic car relocation policy consistently outperforms the standard static policy in realistic instances from the literature. In fact, it reduces the static policy's optimality gap by more than 13% in all of these instances and up to 35% in some.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128227618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Explicit Expansions for Multivariate Diffusions 多元扩散的显式展开式
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2020-12-07 DOI: 10.2139/ssrn.3748893
Xiangwei Wan, Nian Yang
{"title":"Explicit Expansions for Multivariate Diffusions","authors":"Xiangwei Wan, Nian Yang","doi":"10.2139/ssrn.3748893","DOIUrl":"https://doi.org/10.2139/ssrn.3748893","url":null,"abstract":"In this paper, by adopting a quasi-Lamperti transform unitizing the process’ diffusion matrix at the initial time, we provide a new explicit recursive formula to compute the expansion coefficients for the pathwise Taylor expansion method of Li (2013). The quasi-Lamperti transform also allows us to recalculate the expansion coefficients for the Hermite expansion method of Wan and Yang (2020) via the pathwise Taylor expansion method, and proves the equivalence between the methods of Li (2013) and Wan and Yang (2020). Based on the established connection between the delta expansion method of Yang et al. (2019) and the Hermite expansion method of Wan and Yang (2020), we unifies the three expansion methods.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126551605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Bidual Approaches in Risk Representation 风险表示的双重方法
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2020-11-09 DOI: 10.2139/ssrn.3727433
A. Balbás, B. Balbás, Raquel Balbás
{"title":"Bidual Approaches in Risk Representation","authors":"A. Balbás, B. Balbás, Raquel Balbás","doi":"10.2139/ssrn.3727433","DOIUrl":"https://doi.org/10.2139/ssrn.3727433","url":null,"abstract":"Downside and deviation risk measures are becoming more and more important in many disciplines with clear interfaces with Applied Mathematics and Operations Research. Their dual representations have played critical roles in most of their applications (risk management, portfolio selection, pricing and hedging, etc.), but, to the best of our knowledge, bidual representations were never profoundly studied. New linear bidual representations will be provided, and their great capacity to linearize many problems will be proved, with special focus on risk optimization. This is important because there are very tractable necessary and sufficient optimality conditions and resolution algorithms in Linear Programming. Moreover, in the linearization process, one will have to introduce new decision variables providing us with very important information, such as sensitivities with respect to the selected risk measure and sensitivities with respect to the selected model (model risk). The theory will be presented for general Banach spaces, and an illustrative example will be given.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128065899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Data-driven Dynamic Pricing and Ordering with Perishable Inventory in a Changing Environment 变化环境下易腐烂库存的数据驱动动态定价和订购
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2020-08-07 DOI: 10.2139/ssrn.3401188
N. B. Keskin, Yuexing Li, Jing-Sheng Song
{"title":"Data-driven Dynamic Pricing and Ordering with Perishable Inventory in a Changing Environment","authors":"N. B. Keskin, Yuexing Li, Jing-Sheng Song","doi":"10.2139/ssrn.3401188","DOIUrl":"https://doi.org/10.2139/ssrn.3401188","url":null,"abstract":"We consider a retailer that sells a perishable product, making joint pricing and inventory ordering decisions over a finite time horizon of T periods with lost sales. Exploring a real-life data set from a leading supermarket chain, we identify several distinctive challenges faced by such a retailer that have not been jointly studied in the literature: the retailer does not have perfect information on (1) the demand-price relationship, (2) the demand noise distribution, (3) the inventory perishability rate, and (4) how the demand-price relationship changes over time. Furthermore, the demand noise distribution is nonparametric for some products but parametric for others. To tackle these challenges, we design two types of data-driven pricing and ordering (DDPO) policies for the cases of nonparametric and parametric noise distributions. Measuring performance by regret, that is, the profit loss caused by not knowing (1)–(4), we prove that the T-period regret of our DDPO policies are in the order of [Formula: see text] and [Formula: see text] in the cases of nonparametric and parametric noise distributions, respectively. These are the best achievable growth rates of regret in these settings (up to logarithmic terms). Implementing our policies in the context of the aforementioned real-life data set, we show that our approach significantly outperforms the historical decisions made by the supermarket chain. Moreover, we characterize parameter regimes that quantify the relative significance of the changing environment and product perishability. Finally, we extend our model to allow for age-dependent perishability and demand censoring and modify our policies to address these issues. This paper was accepted by David Simchi-Levi, Management Science Special Section on Data-Driven Prescriptive Analytics.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116457323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 25
The Gödel Incompleteness Theorems (1931) by the Axiom of Choice Gödel选择公理的不完备性定理(1931)
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2020-07-12 DOI: 10.2139/ssrn.3649398
Vasil Penchev
{"title":"The Gödel Incompleteness Theorems (1931) by the Axiom of Choice","authors":"Vasil Penchev","doi":"10.2139/ssrn.3649398","DOIUrl":"https://doi.org/10.2139/ssrn.3649398","url":null,"abstract":"Those incompleteness theorems mean the relation of (Peano) arithmeticand (ZFC) set theory, or philosophically, the relation of arithmetical finiteness andactual infinity. The same is managed in the framework of set theory by the axiom ofchoice (respectively, by the equivalent well-ordering \"theorem'). One may discuss thatincompleteness form the viewpoint of set theory by the axiom of choice rather thanthe usual viewpoint meant in the proof of theorems. The logical corollaries from that\"nonstandard\" viewpoint the relation of set theory and arithmetic are demonstrated","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123290765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing 蒙特卡罗期权定价粗糙Bergomi模型的马尔可夫逼近
ERN: Other Econometrics: Mathematical Methods & Programming (Topic) Pub Date : 2020-07-02 DOI: 10.3390/math9050528
Qinwen Zhu, G. Loeper, Wen Chen, Nicolas Langren'e
{"title":"Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing","authors":"Qinwen Zhu, G. Loeper, Wen Chen, Nicolas Langren'e","doi":"10.3390/math9050528","DOIUrl":"https://doi.org/10.3390/math9050528","url":null,"abstract":"The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be well-approximated by the forward-variance Bergomi model with wisely chosen weights and mean-reversion speed parameters (aBergomi), which has the Markovian property. We establish an explicit bound on the L2-error between the respective kernels of these two models, which is explicitly controlled by the number of terms in the aBergomi model. We establish and describe the affine structure of the rBergomi model, and show the convergence of the affine structure of the aBergomi model to the one of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a classical Markovian Monte Carlo simulation scheme for the aBergomi model, which we compare to the hybrid scheme of the rBergomi model.","PeriodicalId":365755,"journal":{"name":"ERN: Other Econometrics: Mathematical Methods & Programming (Topic)","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114877122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
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