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Lévy-Ito models in finance Lévy Ito金融模型
IF 1.6
Probability Surveys Pub Date : 2019-07-19 DOI: 10.1214/21-PS1
G. Bouzianis, L. Hughston, S. Jaimungal, Leandro S'anchez-Betancourt
{"title":"Lévy-Ito models in finance","authors":"G. Bouzianis, L. Hughston, S. Jaimungal, Leandro S'anchez-Betancourt","doi":"10.1214/21-PS1","DOIUrl":"https://doi.org/10.1214/21-PS1","url":null,"abstract":"We propose a class of financial models in which the prices of assets are Levy-Ito processes driven by Brownian motion and a dynamic Poisson random measure. Each such model consists of a pricing kernel, a money market account, and one or more risky assets. The Poisson random measure is associated with an $n$-dimensional Levy process. We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the Levy measure associated with the $n$-dimensional Levy process. The resulting framework is applied to the theory of interest rates and foreign exchange, allowing one to construct new models as well as various generalizations of familiar models.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44436326","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
A unified approach for solving sequential selection problems 求解序列选择问题的统一方法
IF 1.6
Probability Surveys Pub Date : 2019-01-14 DOI: 10.1214/19-ps333
A. Goldenshluger, Y. Malinovsky, A. Zeevi
{"title":"A unified approach for solving sequential selection problems","authors":"A. Goldenshluger, Y. Malinovsky, A. Zeevi","doi":"10.1214/19-ps333","DOIUrl":"https://doi.org/10.1214/19-ps333","url":null,"abstract":"In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47211747","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: 7
Mathematical models of gene expression 基因表达的数学模型
IF 1.6
Probability Surveys Pub Date : 2019-01-01 DOI: 10.1214/19-ps332
Philippe Robert
{"title":"Mathematical models of gene expression","authors":"Philippe Robert","doi":"10.1214/19-ps332","DOIUrl":"https://doi.org/10.1214/19-ps332","url":null,"abstract":"In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49494633","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: 7
Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences 高斯序列二次变分极限定理的充要条件
IF 1.6
Probability Surveys Pub Date : 2019-01-01 DOI: 10.1214/15-PS267
L. Viitasaari
{"title":"Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences","authors":"L. Viitasaari","doi":"10.1214/15-PS267","DOIUrl":"https://doi.org/10.1214/15-PS267","url":null,"abstract":"The quadratic variation of Gaussian processes plays an important rolein both stochastic analysis and in applications such as estimation ofmodel parameters, and for this reason the topic has been extensivelystudied in the literature. In this article we study the convergence ofquadratic sums of general Gaussian sequences. We provide necessary andsufficient conditions for different types of convergence includingconvergence in probability, almost sure convergence, $L^{p}$-convergenceas well as weak convergence. We use a practical and simple approachwhich simplifies the existing methodology considerably. As anapplication, we show how convergence of the quadratic variation of agiven process can be obtained by an appropriate choice of the underlyingsequence.<script type=\"text/javascript\"src=\"//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47084813","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: 12
Poisson approximation 泊松近似
IF 1.6
Probability Surveys Pub Date : 2018-12-31 DOI: 10.1007/978-0-387-22711-5_14
S.Y.Novak
{"title":"Poisson approximation","authors":"S.Y.Novak","doi":"10.1007/978-0-387-22711-5_14","DOIUrl":"https://doi.org/10.1007/978-0-387-22711-5_14","url":null,"abstract":"","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43435812","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
Some algebraic structures in KPZ universality KPZ泛性中的一些代数结构
IF 1.6
Probability Surveys Pub Date : 2018-12-18 DOI: 10.1214/19-ps335
Nikos Zygouras
{"title":"Some algebraic structures in KPZ universality","authors":"Nikos Zygouras","doi":"10.1214/19-ps335","DOIUrl":"https://doi.org/10.1214/19-ps335","url":null,"abstract":"We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is given on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov and we present how these are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct stochastic dynamics on two dimensional arrays called Gelfand-Tsetlin patterns, in ways that couple different one dimensional stochastic processes. The goal of the notes is to expose some of the overarching principles, that have driven a significant number of developments in the field, as a unifying theme.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2018-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47573623","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: 21
Time-uniform Chernoff bounds via nonnegative supermartingales 非负上鞅的时一致Chernoff界
IF 1.6
Probability Surveys Pub Date : 2018-08-09 DOI: 10.1214/18-ps321
Steven R. Howard, Aaditya Ramdas, Jon D. McAuliffe, J. Sekhon
{"title":"Time-uniform Chernoff bounds via nonnegative supermartingales","authors":"Steven R. Howard, Aaditya Ramdas, Jon D. McAuliffe, J. Sekhon","doi":"10.1214/18-ps321","DOIUrl":"https://doi.org/10.1214/18-ps321","url":null,"abstract":"We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960-80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Pe~na; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cram'er-Chernoff method, self-normalized processes, and other parts of the literature.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46546439","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: 115
The Bethe ansatz for the six-vertex and XXZ models: An exposition 六顶点和XXZ模型的Bethe分析:阐述
IF 1.6
Probability Surveys Pub Date : 2016-11-29 DOI: 10.1214/17-PS292
H. Duminil-Copin, Maxime Gagnebin, Matan Harel, I. Manolescu, V. Tassion
{"title":"The Bethe ansatz for the six-vertex and XXZ models: An exposition","authors":"H. Duminil-Copin, Maxime Gagnebin, Matan Harel, I. Manolescu, V. Tassion","doi":"10.1214/17-PS292","DOIUrl":"https://doi.org/10.1214/17-PS292","url":null,"abstract":"In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector (psi) and energy (Lambda), which satisfy (V psi = Lambda psi), where (V) is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights (a= b=1) and (c > 0). We also show that the same vector (psi) satisfies (H psi = E psi), where (H) is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value (E) computed explicitly. \u0000 \u0000Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a \u0000pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on (mathbb{Z}^{2}) with cluster weight (q >4) exhibits a first-order phase transition. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"15 1","pages":"102-130"},"PeriodicalIF":1.6,"publicationDate":"2016-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-PS292","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66071434","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: 22
Equidistribution, uniform distribution: a probabilist's perspective 均衡分布，均匀分布:一个概率学家的观点
IF 1.6
Probability Surveys Pub Date : 2016-10-07 DOI: 10.1214/17-PS295
V. Limic, Nedvzad Limi'c
{"title":"Equidistribution, uniform distribution: a probabilist's perspective","authors":"V. Limic, Nedvzad Limi'c","doi":"10.1214/17-PS295","DOIUrl":"https://doi.org/10.1214/17-PS295","url":null,"abstract":"The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed (mathsf{mod}~1) sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original (p)-multiply equidistributed sequence (k^p, t {mathsf{mod}}~1), (kgeq1) (where (pinmathbb{N}) and (tin[0,1])), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. \u0000 \u0000The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to (infty)-distributed Koksma's numbers (t^k {mathsf{mod}}~1), (kgeq1) (where (tin[1,a]) for some (a>1)), and an important generalization by Niederreiter and Tichy from 1985 are discussed. \u0000 \u0000The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as \u0000certain computer scientists and number theorists. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"15 1","pages":"131-155"},"PeriodicalIF":1.6,"publicationDate":"2016-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66071523","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
On the scaling limits of weakly asymmetric bridges 弱不对称桥梁的标度极限
IF 1.6
Probability Surveys Pub Date : 2016-09-19 DOI: 10.1214/17-PS285
C. Labb'e
{"title":"On the scaling limits of weakly asymmetric bridges","authors":"C. Labb'e","doi":"10.1214/17-PS285","DOIUrl":"https://doi.org/10.1214/17-PS285","url":null,"abstract":"We consider a discrete bridge from ((0,0)) to ((2N,0)) evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order (N^{-alpha}) with (alphain(0,infty)). We provide a classification of the asymptotic behaviours - invariant measure, hydrodynamic limit and fluctuations - of this model according to the value of the parameter (alpha).<script type=\"text/javascript\"src=\"//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"15 1","pages":"156-242"},"PeriodicalIF":1.6,"publicationDate":"2016-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66071747","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: 7
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