Stochastic partial dierential equations whose solutions are probability-measurevalued processes are considered. Measure-valued processes of this type arise naturally as de Finetti measures of innite exchangeable systems of particles and as the solutions for ltering problems. In both these cases, the solution is the conditional distribution of the solution of a stochastic dierential equation. The main result states that, under mild nondegeneracy conditions on the coecients of the stochastic dierential equation, the conditional distribution of its solution charges any open set. Under stronger conditions we show that it is absolutely continuous with respect to Lebesgue measure and its density is positive almost everywhere. As applications we show the existence of a solution of a system of interacting diusions and study the properties of the solution of the nonlinear ltering equation within a framework that allows for the signal noise and the observation noise to be correlated. The work was motivated by a model of asset price determination in which the price is given as a quantile of the valuations of innitely many individual investors. MSC 2010 subject classications: 60H15, 60G09, 60G35, 60J25
{"title":"Conditional distributions, exchangeable particle systems, and stochastic partial differential equations","authors":"D. Crisan, T. Kurtz, Yoonjung Lee","doi":"10.1214/13-AIHP543","DOIUrl":"https://doi.org/10.1214/13-AIHP543","url":null,"abstract":"Stochastic partial dierential equations whose solutions are probability-measurevalued processes are considered. Measure-valued processes of this type arise naturally as de Finetti measures of innite exchangeable systems of particles and as the solutions for ltering problems. In both these cases, the solution is the conditional distribution of the solution of a stochastic dierential equation. The main result states that, under mild nondegeneracy conditions on the coecients of the stochastic dierential equation, the conditional distribution of its solution charges any open set. Under stronger conditions we show that it is absolutely continuous with respect to Lebesgue measure and its density is positive almost everywhere. As applications we show the existence of a solution of a system of interacting diusions and study the properties of the solution of the nonlinear ltering equation within a framework that allows for the signal noise and the observation noise to be correlated. The work was motivated by a model of asset price determination in which the price is given as a quantile of the valuations of innitely many individual investors. MSC 2010 subject classications: 60H15, 60G09, 60G35, 60J25","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"103 1","pages":"946-974"},"PeriodicalIF":1.5,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91005837","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}
Jingsheng Ma, Zhenjie Ren, N. Touzi, Jianfeng Zhang
This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao & Liu [19], this extends the corresponding results collected in Freidlin & Wentzell [18]. However, we use a different line of argument, adapting the PDE method of Fleming [14] and Evans & Ishii [10] to the pathdependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.
{"title":"Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation","authors":"Jingsheng Ma, Zhenjie Ren, N. Touzi, Jianfeng Zhang","doi":"10.1214/15-AIHP678","DOIUrl":"https://doi.org/10.1214/15-AIHP678","url":null,"abstract":"This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao & Liu [19], this extends the corresponding results collected in Freidlin & Wentzell [18]. However, we use a different line of argument, adapting the PDE method of Fleming [14] and Evans & Ishii [10] to the pathdependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"15 1","pages":"1196-1216"},"PeriodicalIF":1.5,"publicationDate":"2014-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88907199","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 prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.
{"title":"Total length of the genealogical tree for quadratic stationary continuous-state branching processes","authors":"Hongwei Bi, Jean-François Delmas","doi":"10.1214/15-AIHP683","DOIUrl":"https://doi.org/10.1214/15-AIHP683","url":null,"abstract":"We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"65 1","pages":"1321-1350"},"PeriodicalIF":1.5,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85301736","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}
Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A1,...,Am of S(X) and checking whether the fraction of points in B that lie in Ak is close to u(Ak) for each k = 1,...,m. We show that if B is any orthonormal basis of X and m is not too large, then, if we randomize the test by applying a random rotation to the sets A1,...,Am, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described.
设X是一个大维有限的实数或复希尔伯特空间d，设S(X)表示X的单位球，设u表示S(X)上的归一化一致测度。对于S(X)的有限子集B，我们可以通过选择划分A1，…来检验它是否近似均匀分布在球上。，Am (S(X))，并检查对于每个k = 1，…，m, B中位于Ak中的点的分数是否接近u(Ak)。我们证明，如果B是X的任意正交基，m不太大，那么，如果我们通过对集合A1，…， a, B通过随机测试的概率接近于1。这个说法与大数定律有关，但不包含在大数定律中。简述了这一事实在量子统计力学中的应用。
{"title":"Any orthonormal basis in high dimension is uniformly distributed over the sphere","authors":"S. Goldstein, J. Lebowitz, R. Tumulka, N. Zanghí","doi":"10.1214/15-AIHP732","DOIUrl":"https://doi.org/10.1214/15-AIHP732","url":null,"abstract":"Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A1,...,Am of S(X) and checking whether the fraction of points in B that lie in Ak is close to u(Ak) for each k = 1,...,m. We show that if B is any orthonormal basis of X and m is not too large, then, if we randomize the test by applying a random rotation to the sets A1,...,Am, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"155 1","pages":"701-717"},"PeriodicalIF":1.5,"publicationDate":"2014-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79625946","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}
Pascal Maillard, Rémi Rhodes, V. Vargas, O. Zeitouni
Dans ce papier, nous initions l’etude des proprietes analytiques du noyau de la chaleur de Liouville. En particulier, nous etablissons des estimees de regularite pour le noyau et nous l’encadrons par des bornes inferieures et superieures non triviales.
{"title":"Liouville heat kernel: Regularity and bounds","authors":"Pascal Maillard, Rémi Rhodes, V. Vargas, O. Zeitouni","doi":"10.1214/15-AIHP676","DOIUrl":"https://doi.org/10.1214/15-AIHP676","url":null,"abstract":"Dans ce papier, nous initions l’etude des proprietes analytiques du noyau de la chaleur de Liouville. En particulier, nous etablissons des estimees de regularite pour le noyau et nous l’encadrons par des bornes inferieures et superieures non triviales.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"98 1","pages":"1281-1320"},"PeriodicalIF":1.5,"publicationDate":"2014-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77027313","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}
Consider a monotone Boolean function f:{0,1}^n to {0,1} and the canonical monotone coupling {eta_p:p in [0,1]} of an element in {0,1}^n chosen according to product measure with intensity p in [0,1]. The random point p in [0,1] where f(eta_p) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majorityand percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.
{"title":"Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?","authors":"Daniel Ahlberg, J. Steif, G. Pete","doi":"10.1214/16-AIHP786","DOIUrl":"https://doi.org/10.1214/16-AIHP786","url":null,"abstract":"Consider a monotone Boolean function f:{0,1}^n to {0,1} and the canonical monotone coupling \u0000{eta_p:p in [0,1]} of an element in {0,1}^n chosen according to product measure with intensity \u0000p in [0,1]. The random point p in [0,1] where f(eta_p) flips from 0 to 1 is often concentrated \u0000near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, \u0000we peer closely into this threshold window and consider, for large n, the limiting distribution (properly \u0000normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. \u0000We determine this distribution for a number of the Boolean functions which are typically studied and pay \u0000particular attention to the functions corresponding to iterated majorityand percolation crossings. It turns out \u0000that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate \u0000probability measure on R arises in this way for some sequence of Boolean functions.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"75 1","pages":"2135-2161"},"PeriodicalIF":1.5,"publicationDate":"2014-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90678978","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 are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.
{"title":"Long time behaviour and stationary regime of memory gradient diffusions","authors":"S. Gadat, Fabien Panloup","doi":"10.1214/12-AIHP536","DOIUrl":"https://doi.org/10.1214/12-AIHP536","url":null,"abstract":"In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"2016 1","pages":"564-601"},"PeriodicalIF":1.5,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86526836","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}
Let {X ij }, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 = µ, E|X 11 − µ| 2 = 1 and E|X 11 | 4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1 n n j=1 (s j − ¯ s)(s j − ¯ s) T and S = 1 n n j=1 s j s T j , where ¯ s = 1 n n j=1 s j and s j = T 1/2 n (X 1j , · · · , X pj) T with (T 1/2 n) 2 = T n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of S and S are different as n → ∞ with p/n approaching a positive constant. Moreover , it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.
设{X ij}， i, j =···为独立同分布(i.i.d)实数随机变量的双数组，其中EX 11 =µ，E|X 11−µ| 2 = 1,E|X 11 | 4 <∞。考虑样本协方差矩阵(有/没有经验定心)S =1 n n j=1 (S j−¯S)(S j−¯S) T和S =1 n n j=1 S j S T j，其中¯S =1 n n j=1 S j和S j= t1 /2 n (X 1j，···，X pj) T with (T 1/2 n) 2 = T n，非随机对称非负定矩阵。证明了当n→∞且p/n趋近于正常数时S和S的特征值统计量的中心极限定理是不同的。此外，还证明了在特征向量的平均行为中没有观察到这种不同的行为。
{"title":"Comparison between two types of large sample covariance matrices","authors":"G. Pan","doi":"10.1214/12-AIHP506","DOIUrl":"https://doi.org/10.1214/12-AIHP506","url":null,"abstract":"Let {X ij }, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 = µ, E|X 11 − µ| 2 = 1 and E|X 11 | 4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1 n n j=1 (s j − ¯ s)(s j − ¯ s) T and S = 1 n n j=1 s j s T j , where ¯ s = 1 n n j=1 s j and s j = T 1/2 n (X 1j , · · · , X pj) T with (T 1/2 n) 2 = T n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of S and S are different as n → ∞ with p/n approaching a positive constant. Moreover , it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"15 1","pages":"655-677"},"PeriodicalIF":1.5,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87010067","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}
Given a two-dimensional fractional multiplicative process (Ft)t∈[0,1] determined by two Hurst exponents H1 and H2, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of [0, 1] by F if and only if H1 = H2.
{"title":"A Uniform Dimension Result for Two-Dimensional Fractional Multiplicative Processes","authors":"Xiong Jin","doi":"10.1214/12-AIHP509","DOIUrl":"https://doi.org/10.1214/12-AIHP509","url":null,"abstract":"Given a two-dimensional fractional multiplicative process (Ft)t∈[0,1] determined by two Hurst exponents H1 and H2, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of [0, 1] by F if and only if H1 = H2.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"97 1","pages":"512-523"},"PeriodicalIF":1.5,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79027230","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}
Thibaut Mastrolia, Dylan Possamai, Anthony R'eveillac
In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. Incidentally , we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .
{"title":"On the Malliavin differentiability of BSDEs","authors":"Thibaut Mastrolia, Dylan Possamai, Anthony R'eveillac","doi":"10.1214/15-AIHP723","DOIUrl":"https://doi.org/10.1214/15-AIHP723","url":null,"abstract":"In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. Incidentally , we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"83 1","pages":"464-492"},"PeriodicalIF":1.5,"publicationDate":"2014-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88907426","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：