Brownian excursion

Short description: Stochastic process

A realization of Brownian Excursion.

In probability theory a Brownian excursion process (BPE) is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1\. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.[1]

## Contents

* 1 Constructions
* 2 Properties
* 3 Connections and applications
* 4 Related processes
* 5 Notes
* 6 References

## Constructions

File:Brownian bridge as a union of excursions.svgBrownian bridge as a union of excursions.

A Brownian excursion process, e, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1\. Alternatively, it is a Brownian bridge process conditioned to be positive. Although note that, since the probability for an unrestricted Brownian bridge to be positive is zero, the conditioning requires care.

Another representation of a Brownian excursion e in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.[2]) is in terms of the last time τ− that W hits zero before time 1 and the first time τ+ that Brownian motion W hits zero after time 1:[2]

{e(t): 0≤t≤1} =d {|W((1−t)τ−+tτ+)|τ+−τ−: 0≤t≤1},

where the square root is due to the square root self-similarity of Wiener process. That is,

aWt/a

is a Wiener process for any fixed constant

a>0

File:Vervaat's transformation.svgVervaat's transformation.

Let τm be the time that a Brownian bridge process W0 achieves its minimum on [0, 1]. Vervaat (1979) shows that

{e(t): 0≤t≤1} =d {W0(τm+tmod1)−W0(τm): 0≤t≤1}.

This is sometimes called Vervaat's transformation.

## Properties

Vervaat's representation of a Brownian excursion has several consequences for various functions of e. In particular:

M+≡sup0≤t≤1e(t) =d sup0≤t≤1W0(t)−inf0≤t≤1W0(t),

(this can also be derived by explicit calculations[3][4]) and

∫01e(t)dt =d ∫01W0(t)dt−inf0≤t≤1W0(t).

The following result holds:[5]

EM+=π/2≈1.25331…,

and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:[5]

EM+2≈1.64493… , Var⁡(M+)≈0.0741337….

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of ∫01e(t)dt. A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984).

Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion W in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of W.

For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII.

## Connections and applications

With probability 1, a Wiener process is continuous, which means the set on which it is non-zero is an open subset of the real line, thus it is the union of countably many Brownian excursions.

The Brownian excursion area

A+≡∫01e(t)dt

arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.[6][7][8][9][10] and the limit distribution of the Betti numbers of certain varieties in cohomology theory.[11] Takacs (1991a) shows that A+ has density

fA+(x)=26x2∑j=1∞vj2/3e−vjU(−56,43;vj) with vj=2|aj|327x2

where aj are the zeros of the Airy function and U is the confluent hypergeometric function. Janson and Louchard (2007) show that

fA+(x)∼726πx2e−6x2 as x->∞,

and

P(A+>x)∼66πxe−6x2 as x->∞.

They also give higher-order expansions in both cases.

Janson (2007) gives moments of A+ and many other area functionals. In particular,

E(A+)=12π2, E(A+2)=512≈0.416666…, Var⁡(A+)=512−π8≈.0239675… .

Brownian excursions also arise in connection with queuing problems,[12] railway traffic,[13][14] and the heights of random rooted binary trees.[15]

## Related processes

* Brownian bridge
* Brownian meander
* reflected Brownian motion
* skew Brownian motion

## Notes

1. ↑ Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)
2. ↑ 2.0 2.1 Itô and McKean (1974, page 75)
3. ↑ Chung (1976)
4. ↑ Kennedy (1976)
5. ↑ 5.0 5.1 Durrett and Iglehart (1977)
6. ↑ Wright, E. M. (1977). "The number of connected sparsely edged graphs". Journal of Graph Theory 1 (4): 317–330. doi:10.1002/jgt.3190010407.
7. ↑ Wright, E. M. (1980). "The number of connected sparsely edged graphs. III. Asymptotic results". Journal of Graph Theory 4 (4): 393–407. doi:10.1002/jgt.3190040409.
8. ↑ Spencer J (1997). "Enumerating graphs and Brownian motion". Communications on Pure and Applied Mathematics 50 (3): 291–294. doi:10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6.
9. ↑ Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys 4: 80–145. doi:10.1214/07-PS104. Bibcode: 2007arXiv0704.2289J.
10. ↑ Flajolet, P.; Louchard, G. (2001). "Analytic variations on the Airy distribution". Algorithmica 31 (3): 361–377. doi:10.1007/s00453-001-0056-0.
11. ↑ Reineke M (2005). "Cohomology of noncommutative Hilbert schemes". Algebras and Representation Theory 8 (4): 541–561. doi:10.1007/s10468-005-8762-y.
12. ↑ Iglehart D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive". The Annals of Probability 2 (4): 608–619. doi:10.1214/aop/1176996607.
13. ↑ Takacs L (1991a). "A Bernoulli excursion and its various applications". Advances in Applied Probability 23 (3): 557–585. doi:10.1017/s0001867800023739.
14. ↑ Takacs L (1991b). "On a probability problem connected with railway traffic". Journal of Applied Mathematics and Stochastic Analysis 4: 263–292. doi:10.1155/S1048953391000011.
15. ↑ Takacs L (1994). "On the Total Heights of Random Rooted Binary Trees". Journal of Combinatorial Theory, Series B 61 (2): 155–166. doi:10.1006/jctb.1994.1041.

## References

* Chung, K. L. (1975). "Maxima in Brownian excursions". Bulletin of the American Mathematical Society 81 (4): 742-745. doi:10.1090/s0002-9904-1975-13852-3. http://projecteuclid.org/euclid.bams/1183537153.
* Chung, K. L. (1976). "Excursions in Brownian motion". Arkiv för Matematik 14 (1): 155-177. doi:10.1007/bf02385832. Bibcode: 1976ArM....14..155C.
* Durrett, Richard T.; Iglehart, Donald L. (1977). "Functionals of Brownian meander and Brownian excursion". Annals of Probability 5 (1): 130-135. doi:10.1214/aop/1176995896.
* Groeneboom, Piet (1983). "The concave majorant of Brownian motion". Annals of Probability 11 (4): 1016-1027. doi:10.1214/aop/1176993450.
* Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields 81: 79-109. doi:10.1007/BF00343738. https://ir.cwi.nl/pub/6435.
* Diffusion Processes and their Sample Paths. Classics in Mathematics (Second printing, corrected ed.). Springer-Verlag, Berlin. 2013. ISBN 978-3540606291.
* Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys 4: 80-145. doi:10.1214/07-ps104. Bibcode: 2007arXiv0704.2289J.
* Janson, Svante; Louchard, Guy (2007). "Tail estimates for the Brownian excursion area and other Brownian areas.". Electronic Journal of Probability 12: 1600-1632. doi:10.1214/ejp.v12-471. Bibcode: 2007arXiv0707.0991J. http://www.emis.de/journals/EJP-ECP/article/view/471.html.
* Kennedy, Douglas P. (1976). "The distribution of the maximum Brownian excursion". Journal of Applied Probability 13 (2): 371-376. doi:10.2307/3212843.
* Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. 1948.
* Louchard, G. (1984). "Kac's formula, Levy's local time and Brownian excursion". Journal of Applied Probability 21 (3): 479-499. doi:10.2307/3213611.
* Pitman, J. W. (1983). "Remarks on the convex minorant of Brownian motion". 5. Birkhauser, Boston. pp. 219-227.
* Revuz, Daniel; Yor, Marc (2004). Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. 293. Springer-Verlag, Berlin. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3.
* Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability 7 (1): 143-149. doi:10.1214/aop/1176995155.

* v
* t
* e

Stochastic processes

Discrete time|

* Bernoulli process
* Branching process
* Chinese restaurant process
* Galton–Watson process
* Independent and identically distributed random variables
* Markov chain
* Moran process
* Random walk
* Loop-erased
* Self-avoiding
* Biased
* Maximal entropy

Continuous time|

* Bessel process
* Birth–death process
* Brownian motion
* Bridge
* Excursion
* Fractional
* Geometric
* Meander
* Cauchy process
* Contact process
* Continuous-time random walk
* Cox process
* Diffusion process
* Empirical process
* Feller process
* Fleming–Viot process
* Gamma process
* Geometric process
* Hunt process
* Interacting particle systems
* Itô diffusion
* Itô process
* Jump diffusion
* Jump process
* Lévy process
* Local time
* Markov additive process
* McKean–Vlasov process
* Ornstein–Uhlenbeck process
* Poisson process
* Compound
* Non-homogeneous
* Point process
* Schramm–Loewner evolution
* Semimartingale
* Sigma-martingale
* Stable process
* Superprocess
* Telegraph process
* Variance gamma process
* Wiener process
* Wiener sausage

Both|

* Branching process
* Galves–Löcherbach model
* Gaussian process
* Hidden Markov model (HMM)
* Markov process
* Martingale
* Differences
* Local
* Sub-
* Super-
* Random dynamical system
* Regenerative process
* Renewal process
* Stochastic chains with memory of variable length
* White noise

Fields and other|

* Dirichlet process
* Gaussian random field
* Gibbs measure
* Hopfield model
* Ising model
* Potts model
* Boolean network
* Markov random field
* Percolation
* Pitman–Yor process
* Point process
* Cox
* Poisson
* Random field
* Random graph

Time series models|

* Autoregressive conditional heteroskedasticity (ARCH) model
* Autoregressive integrated moving average (ARIMA) model
* Autoregressive (AR) model
* Autoregressive–moving-average (ARMA) model
* Generalized autoregressive conditional heteroskedasticity (GARCH) model
* Moving-average (MA) model

Financial models|

* Black–Derman–Toy
* Black–Karasinski
* Black–Scholes
* Chen
* Constant elasticity of variance (CEV)
* Cox–Ingersoll–Ross (CIR)
* Garman–Kohlhagen
* Heath–Jarrow–Morton (HJM)
* Heston
* Ho–Lee
* Hull–White
* LIBOR market
* Rendleman–Bartter
* SABR volatility
* Vašíček
* Wilkie

Actuarial models|

* Bühlmann
* Cramér–Lundberg
* Risk process
* Sparre–Anderson

Queueing models|

* Bulk
* Fluid
* Generalized queueing network
* M/G/1
* M/M/1
* M/M/c

Properties|

* Càdlàg paths
* Continuous
* Continuous paths
* Ergodic
* Exchangeable
* Feller-continuous
* Gauss–Markov
* Markov
* Mixing
* Piecewise deterministic
* Predictable
* Progressively measurable
* Self-similar
* Stationary
* Time-reversible

Limit theorems|

* Central limit theorem
* Donsker's theorem
* Doob's martingale convergence theorems
* Ergodic theorem
* Fisher–Tippett–Gnedenko theorem
* Large deviation principle
* Law of large numbers (weak/strong)
* Law of the iterated logarithm
* Maximal ergodic theorem
* Sanov's theorem

Inequalities|

* Burkholder–Davis–Gundy
* Doob's martingale
* Kunita–Watanabe

Tools|

* Cameron–Martin formula
* Convergence of random variables
* Doléans-Dade exponential
* Doob decomposition theorem
* Doob–Meyer decomposition theorem
* Doob's optional stopping theorem
* Dynkin's formula
* Feynman–Kac formula
* Filtration
* Girsanov theorem
* Infinitesimal generator
* Itô integral
* Itô's lemma
* Karhunen–Loève_theorem
* Kolmogorov continuity theorem
* Kolmogorov extension theorem
* Lévy–Prokhorov metric
* Malliavin calculus
* Martingale representation theorem
* Optional stopping theorem
* Prokhorov's theorem
* Quadratic variation
* Reflection principle
* Skorokhod integral
* Skorokhod's representation theorem
* Skorokhod space
* Snell envelope
* Stochastic differential equation
* Tanaka
* Stopping time
* Stratonovich integral
* Uniform integrability
* Usual hypotheses
* Wiener space
* Classical
* Abstract

Disciplines|

* Actuarial mathematics
* Econometrics
* Ergodic theory
* Extreme value theory (EVT)
* Large deviations theory
* Mathematical finance
* Mathematical statistics
* Probability theory
* Queueing theory
* Renewal theory
* Ruin theory
* Statistics
* System on Chip design
* Stochastic analysis
* Time series analysis
* Machine learning

* List of topics
* Category

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Brownian excursion. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Brownian_excursion&oldid=3977198"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template