Electron. Commun. Probab. 19 (2014), no. 5, 1–5.
DOI: 10.1214/ECP.v19-3215
ISSN: 1083-589X
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
Asymptotics of the probability distributions
of the first hitting times of Bessel processes∗
Yuji Hamana†
Hiroyuki Matsumoto‡
Abstract
The asymptotic behavior of the tail probabilities for the first hitting times of the
Bessel process with arbitrary index is shown without using the explicit expressions
for the distribution function obtained in the authors’ previous works.
Keywords: Bessel process; hitting time; tail probability.
AMS MSC 2010: 60G40.
Submitted to ECP on December 20, 2013, final version accepted on January 29, 2014.
1
Introduction and main results
(ν)
Let Pa be the probability law on the path space W = C([0, ∞); R) of a Bessel
process with index ν ∈ R or dimension δ = 2(ν + 1) starting from a > 0. For w ∈ W we
denote the first hitting time to b > 0 by τb = τb (w):
τb = inf{t > 0; w(t) = b}.
In recent works [4, 5] the authors have shown explicit forms of the distribution
(ν)
function and the density of the distribution of τb under Pa in the case where 0 < b < a.
The other case, which is easier since we do not need to consider a natural boundary,
has been known by Kent [6]. See also Getoor-Sharpe [2].
When ν > 0 and b < a, it is shown in [5] that there exists a positive constant C(ν)
such that
P(ν)
a (τb > t) = 1 −
b
a
2ν
+ C(ν)t−ν + o(t−ν ).
The constant C(ν) may be expressed explicitly and we could treat all the cases. However, we need to consider separately the case where δ is an odd integer and the expression for C(ν) is different from the othere cases. This is because the expression for the
distribution function itself is different.
The aim of this note is to show that the constant C(ν) has the same simple expression
also when δ is an odd integer by considering the asymptotics without using the explicit
expressions for the distribution functions obtained in [5].
(ν)
When a < b and ν > 0, the explicit expressions for Pa (τb > t) have been shown in
[6] and, from his result, it is easily shown that the tail probability decays exponentially.
Hence we concentrate on the case of b < a.
∗ The authors are partially supported by the Grant-in-Aid for scientific Research (C) No. 23540183 and
24540181, Japan Society for the Promotion of Science.
† Kumamoto University, Japan. E-mail: [email protected]
‡ Aoyama Gakuin University, Japan. E-mail: [email protected]
Hitting times of Bessel processes
Theorem 1.1. Let ν > 0 and 0 < b < a. Then, as t → ∞, it holds that
n
b 2ν o
1
2ν
1
−
P(ν)
(t
<
τ
<
∞)
=
b
+ O(t−ν−ε )
b
a
a
Γ(1 + ν)(2t)ν
for any ε ∈ (0,
ν
1+ν ),
where Γ denotes the usual Gamma function.
Theorem 1.2. Let ν < 0 and 0 < b < a. Then, as t → ∞, it holds that
n
b 2|ν| o
1
2|ν|
P(ν)
1−
+ O(t−|ν|−ε )
a (τb > t) = a
a
Γ(1 + |ν|)(2t)|ν|
for any ε ∈ (0,
|ν|
1+|ν| ).
When ν = 0, it is known that
P(0)
a (τb > t) =
2 log(a/b)
+ o((log t)−1 ).
log t
This identity has been discussed in [5] and we omit the details.
2
Proof of Theorem 1.1
We assume ν > 0 in this section. At first we give some lemmas. The first one is
shown by Byczkowski and Ryznar [1].
Lemma 2.1. There exists a constant C , independent of t, such that
−ν
P(ν)
.
a (t < τb < ∞) 5 Ct
Lemma 2.2. If 0 < b < a, one has
P(ν)
a (τb > t) = 1 −
(ν)
for any t > 0, where Ea
the coordinate process.
b 2ν
a
+ E(ν)
a
i
h b 2ν
1{inf 05s5t Rs >b}
Rt
(ν)
is the excpectation with respect to Pa
(2.1)
and {Rs }s=0 denotes
Proof. It is well known that
(ν)
P(ν)
a (τb = ∞) = Pa (inf Rs > b) = 1 −
s=0
b 2ν
a
.
By the Markov property of Bessel processes, we have for t > 0
(ν)
P(ν)
a (τb = ∞) = Pa ( inf Rs > b and inf Rs > b)
05s5t
s=t
(ν)
E(ν)
a [PRt (τb
= ∞)1{inf 05s5t Rs >b} ]
hn
b 2ν o
i
= E(ν)
1−
1{inf 05s5t Rs >b} ,
a
Rt
=
which implies (2.1).
Lemma 2.3. For any a > 0 and p with 0 < p < 1 + ν, it holds that
Γ(1 + ν − p) 1 − a2
Γ(1 + ν − p) 1
−2p
e 2t 5 E(ν)
]5
+ Ct−1−p
a [(Rt )
Γ(1 + ν) (2t)p
Γ(1 + ν) (2t)p
(2.2)
for t = 1, where C is a positive constant independent of t.
ECP 19 (2014), paper 5.
ecp.ejpecp.org
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Hitting times of Bessel processes
Proof. By the explicit expression for the transition density of the Bessel process, we
have
Z ∞
ν
ay 2
2
(ν)
−2p
−2p 1 y
− a +y
Ea [(Rt )
]=
y
t a
0
ye
2t
Iν
t
dy,
where Iν is the modified Bessel function of the first kind with index ν (cf [8]) given by
Iν (z) =
∞
z ν X
2
(z/2)2n
Γ(n + 1)Γ(1 + ν + n)
n=0
(z ∈ R \ (−∞, 0)).
Hence, it is easy to get
−2p
E(ν)
]=
a [(Rt )
∞
1 −a2 /2t X
a2n Γ(n + ν + 1 − p)
e
(2t)p
Γ(n + 1)Γ(1 + n + ν)(2t)n
n=0
and the assertion of the lemma.
Remark 2.4. The moments of Rt for fixed t have explicit expressions by means of the
Whittaker functions (cf. [3], p.709), but it does not seem useful.
We are now in a position to give a complete proof of Theorem 1.1.
Proof of Theorem 1.1. By Lemma 2.2 we have
2ν (ν)
−2ν
−2ν
P(ν)
] − b2ν E(ν)
1{τb 5t} ].
a (t < τb < ∞) = b Ea [(Rt )
a [(Rt )
For the first term we have shown in Lemma 2.3
−2ν
E(ν)
]=
a [(Rt )
1
(1 + O(t−1 )).
Γ(1 + ν)(2t)ν
Hence, if we could show
−2ν
E(ν)
1{τb 5t} ] =
a [(Rt )
for any ε ∈ (0,
ν
ν+1 ),
b 2ν
1 1
+ O ν+ε
ν
Γ(ν + 1)(2t) a
t
(2.3)
we obtain the assertion of the theorem.
For this purpose, we let α ∈ (0,
1
ν+1 ),
choose p satisfying
1
1+ν
<p<
1−α
ν
and let q be such that p−1 + q −1 = 1. We devide the expectation on the right hand side
of (2.3) into the sum of
−2ν
−2ν
I1 = E(ν)
1{τb 5tαq } ] and I2 = E(ν)
1{tαq <τb 5t} ]
a [(Rt )
a [(Rt )
We simply apply the Hölder inequality to I2 . Then we get
−2ν
I2 5 E(ν)
1{tαq <τb <∞} ]
a [(Rt )
n
o1/p n
o1/q
−2νp
(ν) αq
5 E(ν)
[(R
)
]
P
(t
<
τ
<
∞)
.
t
b
a
a
and, by Lemmas 2.1 and 2.3, we see that there exists a constant C1 such that
I2 5 C1 t−ν−αν .
In the following we mean by Ci ’s some constants independent of t.
ECP 19 (2014), paper 5.
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Hitting times of Bessel processes
For I1 , the strong Markov property of Bessel processes implies
Z
tαq
(ν)
Eb [(Rt−s )−2ν ]P(ν)
a (τb ∈ ds) = I11 + I12 ,
I1 =
0
where
tαq
Z
I11 =
0
1
1
P(ν) (τb ∈ ds).
2ν Γ(ν + 1) (t − s)ν a
Since αq < 1, Lemma 2.1 imples
Z
tαq
|I12 | = |I1 − I11 | 5
0
C3
C2
P(ν) (τb ∈ ds) 5 ν+1 .
(t − s)ν+1 a
t
We devide I11 into the sum of
tαq
Z
J1 =
tα
1
2ν Γ(ν
1
P(ν) (τb ∈ ds)
+ 1) (t − s)ν a
and
tα
Z
J2 =
0
1
1
P(ν) (τb ∈ ds).
2ν Γ(ν + 1) (t − s)ν a
For J1 we have by Lemma 2.1
0 5 J1 5
C5
C4
P(ν) (tα < τb < ∞) 5 ν+αν .
(t − tαq )ν a
t
For J2 we have
1
P(ν) (τb 5 tα )
+ 1)(t − tα )ν a
1
1
5
P(ν) (τb < ∞)
Γ(ν + 1)(2t)ν a
(1 − t−(1−α) )ν
b 2ν C6 1
1 + 1−α .
5
ν
Γ(ν + 1)(2t) a
t
J2 5
2ν Γ(ν
On the other hand we have by Lemma 2.1
1
P(ν) (τb 5 tα )
Γ(ν + 1)(2t)ν a
n
o
1
(ν)
(ν) α
=
P
(τ
<
∞)
−
P
(t
5
τ
<
∞)
b
b
a
a
Γ(ν + 1)(2t)ν
2ν
1
b
C7
− ν αν .
=
Γ(ν + 1)(2t)ν a
t t
J2 =
Combining the above estimates, we obtain
−2ν
E(ν)
1{τb 5t} ] =
a [(Rt )
b 2ν
1
1 1
1
1 +
O
+
+
.
Γ(ν + 1)(2t)ν a
tν
tαν
t
t1−α
Since
0 < αν <
ν
<1−α<1
ν+1
and we can choose arbitrary α satifying this condition,
−2ν
E(ν)
1{τb 5t} ] =
a [(Rt )
b 2ν
1 1
+ O ν+ε
ν
Γ(ν + 1)(2t) a
t
ν
holds for any ε ∈ (0, ν+1
).
Now we have shown (2.3) and the assertion of Theorem 1.1.
ECP 19 (2014), paper 5.
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Hitting times of Bessel processes
3
Proof of Theorem 1.2
Theorem 1.2 is easily obtained from Theorem 1.1.
We recall explcit expressions for the Laplace transforms of the distributions of τb :
for ν ∈ R, it is known ([2, 5]) that
−λτb
E(ν)
]
a [e
b ν K (a√2λ)
ν
√
,
=
a Kν (b 2λ)
λ > 0,
where Kν is the modified Bessel function of the second kind. From this identity we
easily obtain for ν > 0
P(−ν)
(τb ∈ dt) =
a
a 2ν
b
P(ν)
a (τb ∈ dt).
Hence we get from Theorem 1.1
P(−ν)
(τb > t) =
a
a 2ν
P(ν)
a (t < τb < ∞)
n
b 2ν o
1
(1 + o(1)).
= a2ν 1 −
a
Γ(1 + ν)(2t)ν
b
Acknowledgement. We thank Professor Yuu Hariya for valuable discussions.
References
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Math. 173, (2006), 19–38. MR-2204460
[2] Getoor, R. K. and Sharpe, M. J.: Excursions of Brownian motion and Bessel processes. Z.
Wahr. Ver. Gebiete 47, (1979), 83–106. MR-0521534
[3] Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series, and Products, 7th ed. Academic
Press, Amsterdam, 2007. xlviii+1171 pp. MR-2360010
[4] Hamana, Y. and Matsumoto, H.: The probability densities of the first hitting times of Bessel
processes. J. Math-for-Industry 4, (2012), 91–95. MR-3072321
[5] Hamana, Y. and Matsumoto, H.: The probability distributions of the first hitting times of
Bessel processes. Trans. AMS 365, (2013), 5237–5257. MR-3074372
[6] Kent, J.T.: Eigenvalue expansions for diffusion hitting times. Z. Wahr. Ver. Gebiete 52, (1980),
309–319. MR-0576891
[7] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, 3rd ed. Springer-Verlag,
Berlin, 1999. xiv+602 pp. MR-1725357
[8] Watson, G. N.: A Treatise on the Theory of Bessel Functions. Reprinted of 2nd ed. Cambridge
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ECP 19 (2014), paper 5.
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