The probability distributions of the first hitting times of Bessel processes

We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover all the cases.


Introduction.
In this article we consider the first hitting time of the Bessel process, which itself is an interesting object and is one of the important tools to study several problems in probability theory. By general theory of one-dimensional diffusion processes, the Laplace transform of the distribution satisfies an eigenvalue problem for the generator and it is given by a ratio of the modified Bessel functions.
Except some special cases it is not easy to invert the Laplace transforms. When the index ν of the Bessel process is a half integer n + 1/2, n ∈ N, the Macdonald function K ν is of a simple form. In this case, it turns that K ν+1 /K ν is represented by the ratio of polynomials. With the help of the partial fraction decomposition, Hamana [8,9] recently has inverted the Laplace transform and applied the results to show the explicit form and the asymptotic behavior of the expected volume of the Wiener sausage for the odd dimensional Brownian motion. The method used in [9] requires some formulae for the zeros of K ν .
The purpose of this paper is to show explicit formulae for the distribution functions and the densities by means of the zeros of the Bessel functions. The results extend the classical ones, for example, due to Ciesielski and Taylor [4], and cover all the cases.
In order to prove the results we represent the ratio of the modified Bessel functions by using contour integrals of functions easier to treat, and invert the Laplace transform. Recently Byczkowski et al [2,3] have given similar but different expressions for the densities of the first hitting times, and applied the results to some study on geometric Brownian motions and hyperbolic Brownian motions. We use the same curve for the contour integral. But, we show a decomposition of the Bessel function ratio and use it, which makes our expression simpler. Moreover, the relation to the classical results in some special cases is clear.
This article is organized as follows. In the next Section 2 we give the main results. After showing two estimates for some ratios of the Bessel functions in Section 3, we prove the main results, Theorems 2.1 and 2.2, in Sections 4 and 5, respectively. Section 6 is devoted to the asymptotic behavior of the tail probability of the first hitting time, which is obtained as an application of the result. In the final Section 7 we show an addition formula for some ratio of the Bessel function, which is guessed from one of our results and is of independent interest.

The first hitting time of the Bessel processes.
For ν ∈ R the one-dimensional diffusion process with infinitesimal generator is called the Bessel process with index ν. If 2ν + 2 is a positive integer, the Bessel process is identical in law with the radial motion of a (2ν +2)-dimensional Brownian motion. Hence, 2ν + 2 is called the dimension of the Bessel process. The classification of boundary points gives the following information. The endpoint ∞ is a natural boundary for any ν ∈ R. For ν ≧ 0, 0 is an entrance and not exit boundary. For −1 < ν < 0, 0 is a regular boundary, which is instantly reflecting. For ν ≦ −1, 0 is an exit but not entrance boundary. For more details, see [10,14] for example.
For a, b ≧ 0 we denote by τ (ν) a,b the first hitting time to b of the Bessel process with index ν starting at a. By general theory of one-dimensional diffusion processes, we can evaluate the Laplace transform of the distribution of τ (ν) a,b by solving an eigenvalue problem. In fact, the function is increasing (decreasing) on [0, b) (resp. (b, ∞)) and satisfies G (ν) u = λu, u(b) = 1.
The following expressions for E[e −λτ (ν) a,b ] is well known (cf. [6,11]): for λ > 0, if b > 0 and ν > −1, if 0 < a ≦ b and ν > −1, if 0 < a ≦ b and ν ≦ −1, if a > 0 and ν < 0, if 0 < b ≦ a and ν ∈ R, Here Γ is the gamma function and I ν and K ν denote modified Bessel functions of the first and the second kinds of order ν, respectively. Both I ν and K ν are the solutions of the modified Bessel differential equation When b > 0 and 2ν + 2 is a positive integer, Ciesielski and Taylor [4] have already shown that, for t > 0 where J µ is the Bessel function of the first kind of order µ and {j ν,k } ∞ k=1 is the increasing sequence of positive zeros of J ν . Hence the density ρ We will use the notation ρ (ν) a,b to denote the probability density function of τ (ν) a,b . While we need to calculate the inverse Laplace transform of the right hand side of (2.1) when 2ν + 2 is not a positive integer, it is possible to obtain it by the same methods as those they used to prove Theorem 1 in [4]. Thus we see that (2.7) is also valid when b > 0 and ν > −1.
In the case a > 0 and ν < 0, we can easily check 2t , by (2.4) and the formula Formulae (2.2) and (2.5) are found in [1], p.398, in the case of ν > 0. Moreover, when 0 < a ≦ b, the following formula is provided: We shall give a proof under more general situation.
Theorem 2.1. Let 0 < a < b. Then, for ν > −1, and, for ν ≦ −1, It should be noted that the result by Ciesielski and Taylor [4] may be obtained by letting a → 0 in (2.8) and using the asymptotic behavior of J ν (z) as z → 0. This theorem immediately shows that, if 0 < a < b and ν > −1, Similar asymptotic result in the case where a = 0 and 2ν + 2 is an integer greater than 2 (Brownian case) was used in [4] to show the law of iterated logarithm for the total time spent by the Bessel process in (0, b) as b ↓ 0. To give our result for the distribution functions of τ (ν) a,b in the case of 0 < b < a, we need to recall some facts about the zeros of the Bessel function K ν . For ν ∈ R we denote by N (ν) the number of zeros of K ν . It is known that N (ν) = |ν| − 1/2 if ν −1/2 is an integer and that N (ν) is the even number closest to |ν|−1/2 otherwise. We remark that N (ν) = 0 if |ν| < 3/2 and N (ν) ≧ 1 if |ν| ≧ 3/2. Each zero, if exists, lies in the half plain {z ∈ C ; Re(z) < 0}, denoted by C − . In this case, we write z ν,1 , z ν,2 , . . . , z ν,N(ν) for the zeros. Since K ν is a solution of (2.6), all zeros of K ν are of multiplicity one by the uniqueness of the solution of ordinary differential equations. This means that all zeros of K ν are distinct. If ν − 1/2 is not an integer, there are no real zeros. For details, see [15], pp.511-513. . ds.
In case of ν = 0, we can not obtain the convenient formula like (2.10) which admits us to apply the Tauberian theorem in a straightforward way. Remark 2.3. It is well known (see, e.g., [14], p.450) that the probability laws of the Bessel processes with different indices are absolutely continuous. Hence formula (2.9) may be deduced from (2.8) and the results in Theorem 2.2 in the case of ν < 0 may be proven from those of ν > 0. Although our proofs work in all the cases, this remark gives a good check for the results.

Some estimates for Bessel functions.
In this section we give two estimates concerning J µ and I µ , which we use in the proof of Theorem 2.1. Throughout this section, we assume that µ > −1 and C i 's are positive constants independent of the variable.
We first recall some facts about the Bessel functions. Let D is the set of points z ∈ C \ {0} with | arg z| < π and {j µ,k } ∞ k=1 denotes the increasing sequence of positive zeros of J µ . It is well known that J µ has no other zeros in D (cf. [12], p.127 and [15], pp.478-484). By virtue of the fact that for m ∈ Z and z ∈ D (see [15], p.75), we see that each −j µ,k is also a zeros of J µ for k ≧ 1. Moreover, in [15], p.77, we find From this formula we see that the zeros of I µ with −π < arg z ≦ π are j µ,k e ±iπ/2 for k ≧ 1.
The following estimates for the Bessel functions of the third kind of order µ, denoted by H (1) µ and H (2) µ , are useful for our purpose. Let δ > 0 be given. Then it is known that, for −π + δ ≦ arg z ≦ 2π − δ, The details are given in [12], p.121 and [15], pp.197-199. The first lemma concerns the exponential growth of the modified Bessel function I µ (z) as |z| → ∞.
Next we show an estimate for the ratio of J µ .
4. The first hitting time in the case of 0 < a < b.
From now on, for a suitable function f , the notation L[f ] implies the Laplace transform of f and the inverse Laplace transform of f is denoted by L −1 [f ]. All formulae concerning Laplace and inverse Laplace transforms can be found in [13]. This section is devoted to a proof of Theorem 2.1. For t > 0 and ν ∈ R let A standard formula shows that Hence it follows from (2.2) and (2.3) that .
Since ϕ µ,a (ze imπ ) = ϕ µ,a (z) for any m ∈ Z, we have that ϕ µ,a is a single-valued holomorphic function on C and an extension of z −µ I µ (az), which means that ϕ µ,a (z) coincides with z −µ I µ (az) for any z ∈ D. Recall that all zeros of I µ are on the imaginary line. Hence zeros of ϕ µ,1 coincide with those of I µ including multiplicities. This yields that the function ϕ µ,a /ϕ µ,1 on C is single-valued, meromorphic and an extension of The singular points of f w µ,c are 0, w and j µ,k e ±iπ/2 for k ≧ 1 and they are all poles of order 1. In order to apply the residue theorem to Ξ(R), we need to consider the zeros of ϕ µ,1 inside C(R) and to take R such that a zero of ϕ µ,1 is not on C(R), but it is sufficient to put because of the following. It is known (cf. [15], p.506) that the large zeros of J µ are given by the asymptotic expansion In particular, for any ε ∈ (0, π/4) there exists an integer n 1 ≧ 1 such that, for n ≧ n 1 Taking the two lemmas in the previous section into account, we take n so large that (4.4) and hold. Then we deduce from the residue theorem It is easy to see that Moreover we have we obtain From (3.1) and (3.2) we deduce and conclude Similarly to (4.6), we have that .
Therefore, summing up the right hand sides of (4.6) and (4.7), we obtain We next prove that Ξ(R n ) tends to 0 as n → ∞. We have which is equal to the summation of the following three integrals: dθ, dθ.
Hence we get Then 0 < δ n < π/2. It follows that Ξ(R n ) is equal to the sum of the following three integrals: By (4.4), we have J µ (R n ) = 0. Applying Lemma 3.1 for η = 1 and R = R n , we obtain This yields that Ξ 1 (R n ) tends to 0 as n → ∞. About Ξ 2 (R n ), it holds that I µ (cR n e iθ e iπ/2 ) I µ (R n e iθ e iπ/2 ) dθ.
we can conclude that Ξ 3 (R n ) tends to 0 as n → ∞ in a similar way as to Ξ 2 (R n ). Accordingly Ξ(R n ) tends to 0 as n → ∞ and the right hand side of (4.8) converges. Therefore we have obtained the following.
and that, for λ > 0 if ν ≦ −1, In order to prove Theorem 2.1, we need to see that, if µ > −1, there exists a positive constant C 11 such that for any k ≧ 1. By formula (3.11) we get 12 /x for some constant C (µ) 12 , which is independent of x. Therefore we obtain (4.14) J .
We are ready to complete our proof of Theorem 2.1. It follows from (4.3) and Therefore we deduce from (4.11) and (4.12) that, if ν > −1, a,b (2b 2 t) for t > 0, we complete our proof of Theorem 2.1. Since (4.13) assures the termwise differentiation with respect to t in (2.8) and (2.9), we obtain the following expression for the densities of τ and that, if ν ≦ −1, .

The first hitting time in the case of 0 < b < a.
This section devoted to a proof of Theorem 2.2. We again use the same notation F (ν) a,b and G (ν) a,b (t) as those in the previous section. Set α = a/b > 1. It follows from (2.5) that, for λ > 0 Since K ν = K −ν for ν ≧ 0, it is sufficient to consider the case where ν ≧ 0. Let ν ≧ 0 and c > 1. We first assume that ν − 1/2 is an integer. In this case, there is a suitable polynomial ψ ν of order ν − 1/2 on C such that ψ ν (0) = 0 and (cf. [12,15]). For example, The function z ν K ν (z) is extended to a entire function and all zeros of ψ ν are the same as those of K ν . For z ∈ C let Then ψ ν,c is a single-valued meromorphic function on C and it holds that for z ∈ D. Therefore, if z ∈ C is not a zero of K ν , we have which implies that K ν (cx)/K ν (x) can be determined uniquely for x < 0 if x is not a zero of K ν . Recall our notation z ν,1 , . . . , z ν,N(ν) for the zeros of K ν . Let w is a point in D with K ν (w) = 0. We take R so large that w and all zeros of K ν are inside C(R), a circle whose center is the origin and radius R.
We set for z ∈ C. The singular points of g w ν,c are 0, w and zeros of K ν , which are all poles of order 1. The residue theorem yields that, if N (ν) = 0, and that, if N (ν) ≧ 1, Res(z ν,j ; g w ν,c ).
By definition of the function g w ν,c , we have If N (ν) ≧ 1, the residue of g w ν,c at z ν,j is equal to for 1 ≦ j ≦ N (ν). Since Lemma 3.1 in [9] gives that, if ψ ν (z 0 ) = 0, .
Θ(R) tends to 0 as R → ∞ since g w ν,c (z) = O(|z| −2 ). Hence we obtain in the case where ν − 1/2 is a non-negative integer. We next consider the case where ν − 1/2 is not an integer and look for a nice expression for K ν (cw)/K ν (w) like (5.5). If ν is not an integer, it is well-known (cf. [15], p.80) that for z ∈ D and m ∈ Z. When ν is an integer, we also have for z ∈ D and m ∈ Z, which is easily seen from for each integer n. Especially, for z ∈ D, we have It follows from these identities that the function K ν (cz)/K ν (z) can not be extended to a meromorphic function on C. For z ∈ D let In order to give a formula for K ν (cw)/K ν (w) like (5.5), we consider the integral of h w ν,c on a suitable contour. However we can not adopt a circle as the contour like (5.2) since h w ν,c can not extend to a meromorphic function on C. Let ε and R be positive numbers with 2ε < R. We set As a contour, we take the curve γ defined by We take R so large and ε so small that w and all zeros of K ν are inside γ. Then, setting The residue theorem yields that, if N (ν) = 0, and that, if N (ν) ≧ 1, Res(z ν,j ; h w ν,c ).
For the integral Π 1 , we have Then, using (5.6) and writing the right hand side by ξ ν , we get Hence, letting γ 0 1 be the line in D defined by γ 0 Here we define three paths as follows: Recall that there is no zero of K ν on the real axis. Then we may apply the Cauchy integral theorem for the integral on the contour consisting of γ 0 1 , γ 1 1 , γ 2 1 and γ 3 1 to obtain Π 3 1 tends to 0 as R → ∞. In fact, noting that ξ ν (xe iθ ) = K ν (xe i(θ+π) ) holds for x > 0, we obtain from (5.10) for |θ| < π/6, which yields for large R and a positive constant C 13 independent of R and θ. Since 0 < θ R < π/6, we see Π 3 1 → 0 as R → ∞. Furthermore (5.11) shows that the function e −(c−1)x ξ ν (cx)/ξ ν (x) is bounded on [ε, ∞) and that Π 2 1 converges as R → ∞. Therefore it holds that In the same way, we can show that for z ∈ D (cf. (5.7)). Note that 1 2πi and recall that the right hand side is L ν,c (x). Then we get x(x + w) dx. (5.12) We will calculate the limit of each term of (5.12) as ε ↓ 0.
The first three terms of the right hand side of (5.12) can be calculated easily. Indeed, Lemma 5.1 yields that the first and the second terms converge to 1/4c ν and that the third term converges to 1/2c ν . By (4.2) and (5.13), we can easily see as x ↓ 0, which has been noted in [3], p.29. Hence the last term of the right hand side of (5.12) converges as ε ↓ 0. Therefore we can conclude Moreover, we can regard z −ν,j as z ν,j for 1 ≦ j ≦ N (ν). Therefore we have proven the following.
(4) If ν − 1/2 is not an integer and |ν| > 3/2, We are ready to complete our proof of Theorem 2.2. We have We need to invert the Laplace transforms of the following functions: The results may be well known (cf. [13]), but we deduce them from the formula At first, put Then we get by (5.15) Next we put Then we obtain from (5.15) Hence we get Now we have shown, for example, for the fourth case where ν − 1/2 is not an integer and |ν| > 3/2, Finally, a simple change of variables from ξ to s given by ξ = (a − b) 2t/s gives us the formula in Theorem 2.2 (4). The other cases are simpler.
6. The tail probability of the first hitting time.
As an application of Theorem 2.2, we show the asymptotic behavior of P (τ (ν) a,b > t) as t → ∞ when 0 < b < a. In Section 2 we showed it when ν = 0 by the Tauberian theorem. In [3], it is shown that, if ν < 0, holds for some constant c ν . It should also be noted that, in [16], Yamazato has discussed on the tail probability in a general framework, and some Bessel processes may be treated. We give an explicit expression for the constant c ν .
To make the statement clear, we define two constants when ν − 1/2 is an integer. Put Moreover we set σ if otherwise.
For |ν| > 1/2 and ν − 1/2 is not an integer we have Then the first term of the right hand side is equal to which is obtained by the Fubini theorem and (6.7). We set Changing variables from x to y given by x √ tu/b = y, we have where 1 A is the indicator function of A. To see the convergence of t |ν| Ψ 0 3 (t; ν) as t → ∞, we need to dominate by an integrable function which is independent of t. We have that (6.8) is equal to Since is bounded on (0, ∞), we have that (6.9) is dominated by a constant multiple of (6.10) and hence (6.10) is bounded by To see that |P (ν) (u)|u −2|ν| is integrable on (0, ∞), we note that for some constant C 14 . Then we get we see that the function given by (6.11) is integrable on (0, ∞) × (0, ∞). Applying the dominated convergence theorem, the Fubini theorem and (5.14), we have that t |ν| Ψ 0 3 (t; ν) tends to Changing variables from u to v given by v = u 2 /2, we have which is equal to 1 2 |ν|+1/2 Γ  When 0 < |ν| < 1/2, it is enough to consider Ψ 0 3 (t; ν) directly. We can easily deduce (6.4) in the same way as Ψ 0 3 (t; ν) for |ν| > 1/2. The calculation is left to the reader.
We are now ready to prove Theorem 6.1. We need only to show Theorem 6.1 in the case of ν = 0.
We lastly consider the case when ν − 1/2 is an integer. Similarly to the case when ν − 1/2 is not an integer, we can deduce the asymptotic behavior for ν > 0 from that for ν < 0. Hence we shall treat only the case of ν < 0. If ν = −1/2, Theorem 2.2 and Lemma 6.4 give Let η ≧ 1 be given and set δ n = Arcsin η R n .
The third term of (7.2) is estimated from above in the same way. Hence we get lim sup n→∞ |Σ(R n )| ≦ C 15 e −2(1−c)η for an arbitrary η ≧ 1 and, therefore, Σ(R n ) tends to 0 as n → ∞.