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On Limit Theorems for the Distribution of the Maximal Element in a Sequence of Random Variables 论随机变量序列中最大元素分布的极限定理
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991854
A. A. Borovkov, E. I. Prokopenko
{"title":"On Limit Theorems for the Distribution of the Maximal Element in a Sequence of Random Variables","authors":"A. A. Borovkov, E. I. Prokopenko","doi":"10.1137/s0040585x97t991854","DOIUrl":"https://doi.org/10.1137/s0040585x97t991854","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 186-204, August 2024. <br/> We study the distribution of the maximal element $overline{xi}_n$ of a sequence of independent random variables $xi_1,dots,xi_n$ and not only for them. The presented approach is more transparent (in our opinion) than the one used before. We consider four classes of distributions with right-unbounded supports and find limit theorems (in an explicit form) of the distribution of $overline{xi}_n$ for them. Earlier, only two classes of right-unbounded distributions were considered, and it was assumed a priori that the normalization of $overline{xi}_n$ is linear; in addition, the components of the normalization (in their explicit form) were unknown. For the two new classes, the required normalization turns our to be nonlinear. Results of this kind are also obtained for four classes of distributions with right-bounded support, which are analogues of the above four right-unbounded distributions (earlier, only the class of distributions with right-bounded support was considered). Some extensions of these results are obtained.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimal Growth Strategies in a Stochastic Market Model with Endogenous Prices 具有内生价格的随机市场模型中的最优增长战略
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991866
M. V. Zhitlukhin
{"title":"Optimal Growth Strategies in a Stochastic Market Model with Endogenous Prices","authors":"M. V. Zhitlukhin","doi":"10.1137/s0040585x97t991866","DOIUrl":"https://doi.org/10.1137/s0040585x97t991866","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 205-216, August 2024. <br/> We consider a stochastic multiagent market model with endogenous asset prices and find a market strategy which cannot be asymptotically outperformed by a single agent. Such a strategy should distribute its capital among the assets proportionally to the conditional expectations of their discounted relative dividend intensities. The main assumption, under which the results are obtained, is that all agents should be small in the sense that actions of an individual agent do not affect the asset prices. The optimal strategy is found as a solution of a linear backward stochastic differential equation.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On an Example of Expectation Evaluation 关于期望评价的一个例子
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991933
A. V. Bulinski
{"title":"On an Example of Expectation Evaluation","authors":"A. V. Bulinski","doi":"10.1137/s0040585x97t991933","DOIUrl":"https://doi.org/10.1137/s0040585x97t991933","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 313-321, August 2024. <br/> We study the distribution of the maximal element $overline{xi}_n$ of a sequence of (possibly) independent random variables $xi_1,dots,xi_n$. A formula for evaluation of a random variable expectation based on a quantile function is considered. This formula is applied to evaluation of the expectation for a nondecreasing function of a random variable transformed via its distribution function. The case of a discontinuous distribution function is the most interesting. As a corollary, we refine an example proposed in the author's previous article [Theory Probab. Appl., 68 (2023), pp. 392--410].","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains 论黄金比率、Tribonacci 数和二阶马尔可夫链的厄尔多斯测度的绝对连续性
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991908
V. L. Kulikov, E. F. Olekhova, V. I. Oseledets
{"title":"On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains","authors":"V. L. Kulikov, E. F. Olekhova, V. I. Oseledets","doi":"10.1137/s0040585x97t991908","DOIUrl":"https://doi.org/10.1137/s0040585x97t991908","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 265-280, August 2024. <br/> We consider a power series at a fixed point $rho in (0.5,1)$, where random coefficients assume a value $0$ or $1$ and form a stationary ergodic aperiodic process. The Erdös measure is the distribution law of such a series. The problem of absolute continuity of the Erdös measure is reduced to the problem of determining when the corresponding hidden Markov chain is a Parry--Markov chain. For the golden ratio and a 1-Markov chains, we give necessary and sufficient conditions for absolute continuity of the Erdös measure and, using Blackwell--Markov chains, provide a new proof that the necessary conditions obtained earlier by Bezhaeva and Oseledets [Theory Probab. Appl., 51 (2007), pp. 28--41] are also sufficient. For tribonacci numbers and 1-Markov chains, we give a new proof of the theorem on singularity of the Erdös measure. For tribonacci numbers and 2-Markov chains, we find only two cases with absolute continuity.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
High Excursion Probabilities for Gaussian Fields on Smooth Manifolds 光滑曲面上高斯场的高偏移概率
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991921
V. I. Piterbarg
{"title":"High Excursion Probabilities for Gaussian Fields on Smooth Manifolds","authors":"V. I. Piterbarg","doi":"10.1137/s0040585x97t991921","DOIUrl":"https://doi.org/10.1137/s0040585x97t991921","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 294-312, August 2024. <br/> Gaussian random fields on finite-dimensional smooth manifolds, whose variance functions reach their maximum values at smooth submanifolds, are considered, and the exact asymptotic behavior of large excursion probabilities is established. It is shown that our conditions on the behavior of the covariation and variance are best possible in the context of the classical Pickands double sum method. Applications of our asymptotic formulas to large deviations of Gaussian vector processes are considered, and some examples are given. This paper continues the previous study of the author with Kobelkov, Rodionov, and Hashorva [J. Math. Sci., 262 (2022), pp. 504--513] which was concerned with Gaussian processes and fields on manifolds with a single point of maximum of the variance.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Monte Carlo Method for Pricing Lookback Type Options in Lévy Models 莱维模型中回溯型期权定价的蒙特卡罗方法
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991891
O. E. Kudryavtsev, A. S. Grechko, I. E. Mamedov
{"title":"Monte Carlo Method for Pricing Lookback Type Options in Lévy Models","authors":"O. E. Kudryavtsev, A. S. Grechko, I. E. Mamedov","doi":"10.1137/s0040585x97t991891","DOIUrl":"https://doi.org/10.1137/s0040585x97t991891","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 243-264, August 2024. <br/> We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener--Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Remark on the Itô Formula 关于伊托公式的评论
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t99188x
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev
{"title":"A Remark on the Itô Formula","authors":"I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev","doi":"10.1137/s0040585x97t99188x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99188x","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 227-242, August 2024. <br/> In the classical Itô formula, we propose replacing the second derivative (understood in the usual sense) by the second derivative in the sense of differentiation of distributions. In particular, we show that this can be done if the first derivative lies in the class $L_{2,mathrm{loc}}(mathbf{R})$. Earlier, Föllmer, Protter, and Shiryayev [Bernoulli, 1 (1995), pp. 149--169] obtained a different form of the last term in the Itô formula under the same conditions.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
300 Years of the Russian Academy of Sciences 俄罗斯科学院成立 300 周年
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991830
A. N. Shiryaev
{"title":"300 Years of the Russian Academy of Sciences","authors":"A. N. Shiryaev","doi":"10.1137/s0040585x97t991830","DOIUrl":"https://doi.org/10.1137/s0040585x97t991830","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 169-172, August 2024. <br/> This year, 2024, the Russian Academy of Sciences celebrates the significant date of the 300th anniversary of its creation. The paper provides brief information about the formation of the Academy and describes its current state.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
In Memory of A. M. Vershik (12.28.1933--02.14.2024) 纪念 A. M. Vershik（1933 年 12 月 28 日-2024 年 2 月 14 日）
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991957
A. I. Bufetov, I. A. Ibragimov, M. A. Lifshits, A. V. Malyutin, F. V. Petrov, N. V. Smorodina, A. N. Shiryaev, Yu. V. Yakubovich
{"title":"In Memory of A. M. Vershik (12.28.1933--02.14.2024)","authors":"A. I. Bufetov, I. A. Ibragimov, M. A. Lifshits, A. V. Malyutin, F. V. Petrov, N. V. Smorodina, A. N. Shiryaev, Yu. V. Yakubovich","doi":"10.1137/s0040585x97t991957","DOIUrl":"https://doi.org/10.1137/s0040585x97t991957","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 331-335, August 2024. <br/> A remembrance of the life and accomplishments of outstanding mathematician Anatolii Moiseevich Vershik, who passed away on February 14, 2024.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance 论连续和的分布与普罗霍罗夫距离的接近性
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991878
A. Yu. Zaitsev
{"title":"On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance","authors":"A. Yu. Zaitsev","doi":"10.1137/s0040585x97t991878","DOIUrl":"https://doi.org/10.1137/s0040585x97t991878","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 217-226, August 2024. <br/> Let $X, X_1,dots, X_n,dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/sqrt{n}$. Then $(X_1+dots+X_n)/sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $pi(,{cdot},,{cdot},)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)&gt;0$ and $c_2(F)&gt;0$ depending only on $F$ such that $pi(F_{(n)}^n, F_{(n)}^{n+1})leqslant c_1(F)/sqrt n$ and $(F^n){A} le (F^{n+1}){A^{c_2(F)}}+c_2(F)/sqrt{n}$, $(F^{n+1}){A} leq (F^n){A^{c_2(F)}}+c_2(F)/sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{varepsilon}$ denotes the $varepsilon$-neighborhood of a set $A$).","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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