{"title":"Special issue dedicated to Alain Bensoussan on the occasion of his 80th birthday: Preface","authors":"R. Buckdahn, Juan Li, S. Peng","doi":"10.3934/puqr.2022010","DOIUrl":"https://doi.org/10.3934/puqr.2022010","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"135 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73766782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a new method to estimate the diffusion function in the jump-diffusion model. First, a threshold reweighted Nadaraya–Watson-type estimator is introduced. Then, we establish asymptotic normality for the estimator and conduct Monte Carlo simulations through two examples to verify the better finite-sampling properties. Finally, our estimator is demonstrated through the actual data of the Shanghai Interbank Offered Rate in China.
{"title":"Threshold reweighted Nadaraya–Watson estimation of jump-diffusion models","authors":"Kunyang Song, Yuping Song, Hanchao Wang","doi":"10.3934/puqr.2022003","DOIUrl":"https://doi.org/10.3934/puqr.2022003","url":null,"abstract":"In this paper, we propose a new method to estimate the diffusion function in the jump-diffusion model. First, a threshold reweighted Nadaraya–Watson-type estimator is introduced. Then, we establish asymptotic normality for the estimator and conduct Monte Carlo simulations through two examples to verify the better finite-sampling properties. Finally, our estimator is demonstrated through the actual data of the Shanghai Interbank Offered Rate in China.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83451965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p style='text-indent:20px;'>Standard mathematical economics studies the production, exchange, and consumption of goods “<i>provided with units of measurement</i>,” as in physics, in order to be enumerated, quantified, added, etc. Therefore, “baskets of goods,” which should describe subsets of goods, are mathematically represented as commodity vectors of a vector space, linear combination of units of goods, evaluated by prices, which are linear numerical functions. Therefore, in this sense, mathematical economics is a branch of physics.</p><p style='text-indent:20px;'>However, economics, and many other domains of life sciences, investigate also what will be called <i>entities</i>, defining <i>elements deprived of units of measure</i>, which thus cannot be enumerated.</p><p style='text-indent:20px;'>(1) Denoting by <inline-formula><tex-math id="M1">begin{document}$X$end{document}</tex-math></inline-formula> the set of entities <inline-formula><tex-math id="M2">begin{document}$x in X$end{document}</tex-math></inline-formula> <i>deprived of units of measurement</i>, a “basket of goods” is actually a <i>subset</i> <inline-formula><tex-math id="M3">begin{document}$K subset X$end{document}</tex-math></inline-formula> of the set entities, i.e., an element of the “<i>hyperset</i>” <inline-formula><tex-math id="M4">begin{document}${cal{P}}(X)$end{document}</tex-math></inline-formula>, the family of subsets of <inline-formula><tex-math id="M5">begin{document}$X$end{document}</tex-math></inline-formula>, and no longer a commodity vector of the vector space of commodities;</p><p style='text-indent:20px;'>(2) Entities can be “gathered” instead of being “added”;</p><p style='text-indent:20px;'>(3) Entities can still be evaluated by a <i>family</i> of functions <inline-formula><tex-math id="M6">begin{document}$A: x in X mapsto A(x) in mathbb{R}$end{document}</tex-math></inline-formula> regarded as a “valuators,” in lieu and place of linear “prices” evaluating the units of economic goods.</p><p style='text-indent:20px;'>(4) Subsets of entities can be evaluated by an “<i>interval of values</i>” <i>between two extremal ones</i>, the minimum and the maximum, instead of the sum of values of units of goods weighted by their quantities.</p><p style='text-indent:20px;'>Life sciences dealing with intertwined relations among many combinations of entities, hypersets offer metaphors of “Lamarckian complexity” that keeps us away from binary relations, graphs of functions, and set-valued maps, to focus our attention on “<i>multinary relations</i>” between families of hypersets. Even deprived of units of measurement, these “proletarian” entities still enjoy enough properties for this pauperization to be mathematically consistent.</p><p style='text-indent:20px;'>This is the object of this <i>extremal manifesto</i>: <i>in economics and other domains of life sciences, vector spaces should yield their imperial status of “state space” to hypersets and linear prices to hypervaluato
Standard mathematical economics studies the production, exchange, and consumption of goods “provided with units of measurement,” as in physics, in order to be enumerated, quantified, added, etc. Therefore, “baskets of goods,” which should describe subsets of goods, are mathematically represented as commodity vectors of a vector space, linear combination of units of goods, evaluated by prices, which are linear numerical functions. Therefore, in this sense, mathematical economics is a branch of physics.However, economics, and many other domains of life sciences, investigate also what will be called entities, defining elements deprived of units of measure, which thus cannot be enumerated.(1) Denoting by begin{document}$X$end{document} the set of entities begin{document}$x in X$end{document} deprived of units of measurement, a “basket of goods” is actually a subset begin{document}$K subset X$end{document} of the set entities, i.e., an element of the “hyperset” begin{document}${cal{P}}(X)$end{document}, the family of subsets of begin{document}$X$end{document}, and no longer a commodity vector of the vector space of commodities;(2) Entities can be “gathered” instead of being “added”;(3) Entities can still be evaluated by a family of functions begin{document}$A: x in X mapsto A(x) in mathbb{R}$end{document} regarded as a “valuators,” in lieu and place of linear “prices” evaluating the units of economic goods.(4) Subsets of entities can be evaluated by an “interval of values” between two extremal ones, the minimum and the maximum, instead of the sum of values of units of goods weighted by their quantities.Life sciences dealing with intertwined relations among many combinations of entities, hypersets offer metaphors of “Lamarckian complexity” that keeps us away from binary relations, graphs of functions, and set-valued maps, to focus our attention on “multinary relations” between families of hypersets. Even deprived of units of measurement, these “proletarian” entities still enjoy enough properties for this pauperization to be mathematically consistent.This is the object of this extremal manifesto: in economics and other domains of life sciences, vector spaces should yield their imperial status of “state space” to hypersets and linear prices to hypervaluators.We no longer have to add goods which can only be gathered, prices do not have to be linear, although it costs some effort to deprive oneself of the powerful and luxurious charms of convex and linear functional analysis motivated by physics.These sacrifices concern only economics and other fields of life science, since physicists deal with experimental observations of objects endowed with unit of measurement by adequate processes of measurement. They can happily live in vector spaces without any guilt. This is not the case of life scientists, who have mainly history to support and validate their observations, with, sometimes, the privilege to statistically measure the frequency of some of them.
{"title":"The value does not exist! A motivation for extremal analysis","authors":"J. Aubin, H. Frankowska","doi":"10.3934/puqr.2022013","DOIUrl":"https://doi.org/10.3934/puqr.2022013","url":null,"abstract":"<p style='text-indent:20px;'>Standard mathematical economics studies the production, exchange, and consumption of goods “<i>provided with units of measurement</i>,” as in physics, in order to be enumerated, quantified, added, etc. Therefore, “baskets of goods,” which should describe subsets of goods, are mathematically represented as commodity vectors of a vector space, linear combination of units of goods, evaluated by prices, which are linear numerical functions. Therefore, in this sense, mathematical economics is a branch of physics.</p><p style='text-indent:20px;'>However, economics, and many other domains of life sciences, investigate also what will be called <i>entities</i>, defining <i>elements deprived of units of measure</i>, which thus cannot be enumerated.</p><p style='text-indent:20px;'>(1) Denoting by <inline-formula><tex-math id=\"M1\">begin{document}$X$end{document}</tex-math></inline-formula> the set of entities <inline-formula><tex-math id=\"M2\">begin{document}$x in X$end{document}</tex-math></inline-formula> <i>deprived of units of measurement</i>, a “basket of goods” is actually a <i>subset</i> <inline-formula><tex-math id=\"M3\">begin{document}$K subset X$end{document}</tex-math></inline-formula> of the set entities, i.e., an element of the “<i>hyperset</i>” <inline-formula><tex-math id=\"M4\">begin{document}${cal{P}}(X)$end{document}</tex-math></inline-formula>, the family of subsets of <inline-formula><tex-math id=\"M5\">begin{document}$X$end{document}</tex-math></inline-formula>, and no longer a commodity vector of the vector space of commodities;</p><p style='text-indent:20px;'>(2) Entities can be “gathered” instead of being “added”;</p><p style='text-indent:20px;'>(3) Entities can still be evaluated by a <i>family</i> of functions <inline-formula><tex-math id=\"M6\">begin{document}$A: x in X mapsto A(x) in mathbb{R}$end{document}</tex-math></inline-formula> regarded as a “valuators,” in lieu and place of linear “prices” evaluating the units of economic goods.</p><p style='text-indent:20px;'>(4) Subsets of entities can be evaluated by an “<i>interval of values</i>” <i>between two extremal ones</i>, the minimum and the maximum, instead of the sum of values of units of goods weighted by their quantities.</p><p style='text-indent:20px;'>Life sciences dealing with intertwined relations among many combinations of entities, hypersets offer metaphors of “Lamarckian complexity” that keeps us away from binary relations, graphs of functions, and set-valued maps, to focus our attention on “<i>multinary relations</i>” between families of hypersets. Even deprived of units of measurement, these “proletarian” entities still enjoy enough properties for this pauperization to be mathematically consistent.</p><p style='text-indent:20px;'>This is the object of this <i>extremal manifesto</i>: <i>in economics and other domains of life sciences, vector spaces should yield their imperial status of “state space” to hypersets and linear prices to hypervaluato","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"331 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80523887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new Hartman–Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.
{"title":"On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations","authors":"Xiao-Qun Guo, Shan Li, Xinpeng Li","doi":"10.3934/puqr.2022001","DOIUrl":"https://doi.org/10.3934/puqr.2022001","url":null,"abstract":"<p style='text-indent:20px;'>A new Hartman–Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.</p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"5 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81160046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by various mean-field type linear-quadratic (MF-LQ, for short) multi-level Stackelberg games, we propose a kind of multi-level self-similar randomized domination-monotonicity structures. When the coefficients of a class of mean-field type forward-backward stochastic differential equations (MF-FBSDEs, for short) satisfy this kind of structures, we prove the existence, the uniqueness, an estimate and the continuous dependence on the coefficients of solutions. Further, the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games, respectively.
{"title":"Mean-field type FBSDEs in a domination-monotonicity framework and LQ multi-level Stackelberg games","authors":"Ran Tian, Zhiyong Yu","doi":"10.3934/puqr.2022014","DOIUrl":"https://doi.org/10.3934/puqr.2022014","url":null,"abstract":"Motivated by various mean-field type linear-quadratic (MF-LQ, for short) multi-level Stackelberg games, we propose a kind of multi-level self-similar randomized domination-monotonicity structures. When the coefficients of a class of mean-field type forward-backward stochastic differential equations (MF-FBSDEs, for short) satisfy this kind of structures, we prove the existence, the uniqueness, an estimate and the continuous dependence on the coefficients of solutions. Further, the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games, respectively.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"41 5 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83208632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"RBSDEs with optional barriers: monotone approximation","authors":"S. Bouhadou, A. Hilbert, Y. Ouknine","doi":"10.3934/puqr.2022005","DOIUrl":"https://doi.org/10.3934/puqr.2022005","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"85 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76047746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The impact of a “quadratic gradient” term in a system of Schrödinger–Maxwell equations","authors":"L. Boccardo","doi":"10.3934/puqr.2022016","DOIUrl":"https://doi.org/10.3934/puqr.2022016","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"57 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81307519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. Closed, not necessarily convex, constraints are imposed on strategies. The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation (BSDE). Due to the stochastic market environment, the solution to this BSDE is unbounded and thereby the BMO argument breaks down. After establishing the martingale optimality criterion, by delicately selecting Lyapunov functions, the verification theorem is ultimately obtained. Besides, several examples and numerical simulations for the optimal strategies are provided and illustrated.
{"title":"Optimal consumption and portfolio selection with Epstein–Zin utility under general constraints","authors":"Zixin Feng, D. Tian","doi":"10.3934/puqr.2023012","DOIUrl":"https://doi.org/10.3934/puqr.2023012","url":null,"abstract":"The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. Closed, not necessarily convex, constraints are imposed on strategies. The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation (BSDE). Due to the stochastic market environment, the solution to this BSDE is unbounded and thereby the BMO argument breaks down. After establishing the martingale optimality criterion, by delicately selecting Lyapunov functions, the verification theorem is ultimately obtained. Besides, several examples and numerical simulations for the optimal strategies are provided and illustrated.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"29 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87748516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.
{"title":"Multi-patch multi-group epidemic model with varying infectivity","authors":"R. Forien, G. Pang, 'Etienne Pardoux","doi":"10.3934/puqr.2022019","DOIUrl":"https://doi.org/10.3934/puqr.2022019","url":null,"abstract":"This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"99 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80963856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that “full-bang” control is optimal in a problem which combines features of (i) sequential least-squares estimation with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded control of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary stopping. We develop also the optimal filtering and stopping rules in this context.
{"title":"A sequential estimation problem with control and discretionary stopping","authors":"Erik Ekstrom, I. Karatzas","doi":"10.3934/puqr.2022011","DOIUrl":"https://doi.org/10.3934/puqr.2022011","url":null,"abstract":"<p style='text-indent:20px;'>We show that “full-bang” control is optimal in a problem which combines features of (i) sequential least-squares <i>estimation</i> with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded <i>control</i> of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary <i>stopping</i>. We develop also the optimal filtering and stopping rules in this context.</p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"86 19 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84012613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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