The probability distributions of the first hitting times of Bessel processes

Hamana, Yuji; Matsumoto, Hiroyuki

doi:10.1090/s0002-9947-2013-05799-6

Cited by 55 publications

(72 citation statements)

References 21 publications

(30 reference statements)

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“…Technical complications appear even for µ = −1/2 (see [16]), it is when the Bessel process corresponds to the one-dimensional Brownian motion. Our aim is to provide asymptotics and uniform estimates of the exit time density in whole range of index µ ∈ R. Analogous results are already known in case x > a [5,10,17], but are obtained by usage of completely different methods. Note that presented in the paper result hold, as a special case, for the first exit time of the Brownian motion from a ball and, up to the knowledge of the author, they were not known before.…”

Section: Introductionmentioning

confidence: 79%

Exit times densities of the Bessel process

Serafin

2017

Proc. Amer. Math. Soc.

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We examine the density functions of the first exit times of the Bessel process from the intervals [0, 1) and (0, 1). First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n-dimensional Brownian motion from a ball.2010 Mathematics Subject Classification. 60J60, 60J65.

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Section: Introductionmentioning

confidence: 79%

Exit times densities of the Bessel process

Serafin

2017

Proc. Amer. Math. Soc.

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“…In this paper we primarily focus on the space‐time distribution of Brownian hitting of a bounded Borel set

A

{boldR}^{d}

expressed as

Q_{A} false(boldx, d t d ξ false) = P_{boldx} false[B_{σ false(A false)} \in d ξ, σ_{A} \in d t false] false(t > 0, d ξ \subset \partial A false) .

Here

P_{x}

denotes the law of a standard Brownian motion

B_{t}

started at

boldx

σ_{A}

(or

σ (A)

) the first hitting time of

A

B_{t}

, namely

σ_{A} = inf false{t > 0 : B_{t} \in A false}

, and

\partial A

the Euclidian boundary of

A

. When

A

is a disc (

d = 2

) or a ball (

d ⩾ 3

), the distribution

P_{boldx} false[σ_{A} < t false]

or its density

P_{boldx} false[σ_{A} \in d t false] / d t

are investigated by several recent works seeking the asymptotic behavior of them for

x \notin A

t \to \infty

. For general

A

the asymptotic form of the distribution

…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…Here P x denotes the law of a standard Brownian motion B t started at x, σ A (or σ(A)) the first hitting time of A by B t , namely σ A = inf{t > 0 : B t ∈ A}, and ∂A the Euclidian boundary of A. When A is a disc (d = 2) or a ball (d 3), the distribution P x [σ A < t] or its density P x [σ A ∈ dt]/dt are investigated by several recent works [5,11,24,26] seeking the asymptotic behavior of them for x / ∈ A as t → ∞. For general A the asymptotic form of the distribution P x [σ A < t] is given by Hunt [12] for d = 2 and by Joffe [15] and Spitzer [22] for d 3.…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

Uchiyama

2017

Proc. London Math. Soc.

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The space‐time distribution, QAfalse(boldx,dt0.16emdξfalse) say, of Brownian hitting of a bounded Borel set A of boldRd is studied. We derive the asymptotic form of the leading term of the time‐derivative QAfalse(boldx,dt0.16emdξfalse)/dt for each d=2,3,…, valid uniformly with respect to the starting point boldx of the Brownian motion, which result significantly extends the classical ones for QAfalse(boldx,dt0.16emdξfalse) itself by Hunt (d=2), Joffe and Spitzer (d⩾3). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

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“…See [7,9] and the references therein for the Laplace forms and related topics. Recently, Byczkowski and Ryznar [3], Uchiyama [11] and the authors of the present paper [4,5,6] have studied the explicit expressions and the asymptotics of the densities themselves and the tail probabilities.…”

Section: Introductionmentioning

confidence: 92%

Hitting times to spheres of Brownian motions with and without drifts

Hamana

Matsumoto

2016

Proc. Amer. Math. Soc.

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Abstract. Explicit formulae for the densities of the first hitting times to the sphere of Brownian motions with drifts are given. We need to consider the joint distributions of the first hitting times to the sphere and the hitting positions of the standard Brownian motion and explicit expression for their Laplace transforms are given, which are different from the known formulae in the literature and are of independnt interest.

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The probability distributions of the first hitting times of Bessel processes

Abstract: Abstract. We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions by means of the zeros of the Bessel functions. The resulting formula is simpler and easier to treat than the corresponding one which has been already obtained.

Cited by 55 publications

References 21 publications

Exit times densities of the Bessel process

Exit times densities of the Bessel process

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

Hitting times to spheres of Brownian motions with and without drifts

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