The Experts below are selected from a list of 4551 Experts worldwide ranked by ideXlab platform
Marc Yor - One of the best experts on this subject based on the ideXlab platform.
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Some explicit formulas for the Brownian Bridge, Brownian meander and Bessel process under uniform sampling
ESAIM: Probability and Statistics, 2015Co-Authors: Mathieu Rosenbaum, Marc YorAbstract:We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian Bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian Bridge together with various relationships between the laws of these four processes.
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On the law of a triplet associated with the pseudo-Brownian Bridge
Lecture Notes in Mathematics, 2014Co-Authors: Mathieu Rosenbaum, Marc YorAbstract:We identify the distribution of a natural triplet associated with the pseudo-Brownian Bridge. In particular, for B a Brownian motion and T 1 its first hitting time of the level one, this remarkable law allows us to understand some properties of the process \((B_{\mathit{uT}_{1}}/\sqrt{T_{1}},\ u \leq 1)\) under uniform random sampling, a study started in (Elie, Rosenbaum, and Yor, On the expectation of normalized Brownian functionals up to first hitting times, Preprint, arXiv:1310.1181, 2013).
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On the law of a triplet associated with the pseudo-Brownian Bridge
2013Co-Authors: Mathieu Rosenbaum, Marc YorAbstract:We identify the distribution of a natural triplet associated with the pseudo-Brownian Bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1},~u\leq 1)$ under uniform random sampling.
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A remark about the norm of a Brownian Bridge
Statistics & Probability Letters, 2004Co-Authors: Marc Yor, Lorenzo ZambottiAbstract:Abstract We prove that the law of the euclidean norm of an n-dimensional Brownian Bridge is, in general, only equivalent and not equal to the law of a n-dimensional Bessel Bridge and we compute explicitly the mutual density. Relations with Bessel processes with drifts are also discussed.
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On the distribution of ranked heights of excursions of a Brownian Bridge
The Annals of Probability, 2001Co-Authors: Jim Pitman, Marc YorAbstract:The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian Bridge (B br t , 0 ≤ t x) = e -2x2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian Bridge (|B br t |, 0 ≤ t ≤ 1) is given by a modification of the known θ-function series for the density of M br 1 = sup 0≤t≤1 |B br t |. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized Bridge of a self-similar recurrent Markov process.
Shoujiang Zhao - One of the best experts on this subject based on the ideXlab platform.
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On the large deviation principle for maximum likelihood estimator of α-Brownian Bridge
Statistics & Probability Letters, 2018Co-Authors: Shoujiang Zhao, Qiaojing Liu, Ting ChenAbstract:Abstract We consider the large deviation principle for maximum likelihood estimator of α -Brownian Bridge. The full large deviation with explicit rate function is obtained in the case of non-Gaussian limit distribution by local large deviation principle and exponential tightness.
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Large deviation expansion for maximum-likelihood estimator of α −Brownian Bridge
Communications in Statistics-theory and Methods, 2016Co-Authors: Shoujiang Zhao, Ting ChenAbstract:ABSTRACTWe studied the large deviation expansion for maximum-likelihood estimator of α −Brownian Bridge by using change of measure and time-varying change of probability. The full expansion provided a useful tool to approximate the tail probability of maximum-likelihood estimator.
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Sharp Large Deviation for the Energy of -Brownian Bridge
International Journal of Stochastic Analysis, 2013Co-Authors: Shoujiang Zhao, Qiaojing Liu, Fuxiang Liu, Hong YinAbstract:We study the sharp large deviation for the energy of -Brownian Bridge. The full expansion of the tail probability for energy is obtained by the change of measure.
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Sharp large deviations for the log-likelihood ratio of an α-Brownian Bridge
Statistics & Probability Letters, 2013Co-Authors: Shoujiang Zhao, Yanping ZhouAbstract:We studied the sharp large deviations for the log-likelihood ratio of an α-Brownian Bridge. The full expansion of the tail probability for the log-likelihood ratio was obtained by using a change of measure.
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Large deviations for parameter estimators of α-Brownian Bridge
Journal of Statistical Planning and Inference, 2012Co-Authors: Shoujiang Zhao, Qiaojing LiuAbstract:Abstract We study large deviations for estimator, quadratic functionals and score function of α - Brownian Bridge. Large deviation in the non-steepness case with explicit rate function is obtained by using change of measure.
Jim Pitman - One of the best experts on this subject based on the ideXlab platform.
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the slepian zero set and Brownian Bridge embedded in Brownian motion by a spacetime shift
Electronic Journal of Probability, 2015Co-Authors: Jim Pitman, Wenpin TangAbstract:This paper is concerned with various aspects of the Slepian process (Bt+1 Bt;t 0) derived from one-dimensional standard Brownian motion (Bt;t 0). In particular, we offer an analysis of the local structure of the Slepian zero setft : Bt+1 = Btg, including a path decomposition of the Slepian process for 0 t 1. We also establish the existence of a random time T 0 such that the process (BT+u BT;0 u 1) is standard Brownian Bridge.
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two recursive decompositions of Brownian Bridge related to the asymptotics of random mappings
Lecture Notes in Mathematics, 2006Co-Authors: David Aldous, Jim PitmanAbstract:Author(s): Aldous, D; Pitman, J | Abstract: Aldous and Pitman (1994) studied asymptotic distributions as n → ∞, of various functional of a uniform random mapping of the set {1,..., n}, by constructing a mapping-walk and showing these random walks converge weakly to a reflecting Brownian Bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian Bridge, each defined by cutting the path of the Bridge at an increasing sequence of recursively defined random times in the zero set of the Bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the Bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel Bridges, and characterize the distributions of various path fragments involved, using the Levy-lto theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index α ∈ (0,1).
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Two recursive decompositions of Brownian Bridge
arXiv: Probability, 2004Co-Authors: David Aldous, Jim PitmanAbstract:Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian Bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian Bridge, each defined by cutting the path of the Bridge at an increasing sequence of recursively defined random times in the zero set of the Bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the Bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel Bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1.
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On the distribution of ranked heights of excursions of a Brownian Bridge
The Annals of Probability, 2001Co-Authors: Jim Pitman, Marc YorAbstract:The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian Bridge (B br t , 0 ≤ t x) = e -2x2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian Bridge (|B br t |, 0 ≤ t ≤ 1) is given by a modification of the known θ-function series for the density of M br 1 = sup 0≤t≤1 |B br t |. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized Bridge of a self-similar recurrent Markov process.
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on the distribution of ranked heights of excursions of a Brownian Bridge
Annals of Probability, 2001Co-Authors: Jim Pitman, Marc YorAbstract:The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian Bridge $(B^{\mathrm{br}}_t, 0 \le t \le 1)$ is described. The height $M^{\mathrm{br} +}_j$of the $j$th highest maximum over a positive excursion of the Bridge has the same distribution as $M^{\mathrm{br} +}_1 /j$, where the distribution of $M^{\mathrm{br} +}_1 = \sup_{0 \le t \le 1} B^{\mathrm{br}}_t$ is given by Levy’s formula $P(M^{\mathrm{br} +}_1 > x) = e^{−2x^{2}}$. The probability density of the height $M^{\mathrm{br}}_j$ of the $j$th highest maximum of excursions of the reflecting Brownian Bridge $(|B^{\mathrm{br}}_t|, 0 \le t \le 1)$ is given by a modification of the known $\theta$-function series for the density of $M^{\mathrm{br}}_1 = \sup_{0 \le t \le 1} |B^{\mathrm{br}}_t|$. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized Bridge of a self-similar recurrent Markov process.
Alessandro Milazzo - One of the best experts on this subject based on the ideXlab platform.
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optimal stopping for the exponential of a Brownian Bridge
Journal of Applied Probability, 2020Co-Authors: Tiziano De Angelis, Alessandro MilazzoAbstract:We study the problem of stopping a Brownian Bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian Bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian Bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.
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Optimal stopping for the exponential of a Brownian Bridge
Journal of Applied Probability, 2020Co-Authors: Tiziano De Angelis, Alessandro MilazzoAbstract:In this paper we study the problem of stopping a Brownian Bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le \tau\le 1}\mathsf{E}[\mathrm{e}^{X_\tau}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian Bridge. Instead, we develop techniques that use pathwise properties of the Brownian Bridge and martingale methods of optimal stopping theory in order to find the optimal stopping rule and to show regularity of the value function.
Qiaojing Liu - One of the best experts on this subject based on the ideXlab platform.
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On the large deviation principle for maximum likelihood estimator of α-Brownian Bridge
Statistics & Probability Letters, 2018Co-Authors: Shoujiang Zhao, Qiaojing Liu, Ting ChenAbstract:Abstract We consider the large deviation principle for maximum likelihood estimator of α -Brownian Bridge. The full large deviation with explicit rate function is obtained in the case of non-Gaussian limit distribution by local large deviation principle and exponential tightness.
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Sharp Large Deviation for the Energy of -Brownian Bridge
International Journal of Stochastic Analysis, 2013Co-Authors: Shoujiang Zhao, Qiaojing Liu, Fuxiang Liu, Hong YinAbstract:We study the sharp large deviation for the energy of -Brownian Bridge. The full expansion of the tail probability for energy is obtained by the change of measure.
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Large deviations for parameter estimators of α-Brownian Bridge
Journal of Statistical Planning and Inference, 2012Co-Authors: Shoujiang Zhao, Qiaojing LiuAbstract:Abstract We study large deviations for estimator, quadratic functionals and score function of α - Brownian Bridge. Large deviation in the non-steepness case with explicit rate function is obtained by using change of measure.
