This paper concerns the first passage times to a point a > 0, denoted by σ a , of Bessel processes. We are interested in the case when the process starts at x > a and compute the densities of the distributions of σ a to obtain the exact asymptotic forms of them as t → ∞ that are valid uniformly in x > a for every order of Bessel process. 1
Introduction and main resultsThis paper concerns the first passage times to a point a ≥ 0, denoted by σ a , of Bessel processes of order ν ∈ R. We are interested in the case when the process starts at x > a and compute the densities of the distributions of σ a to obtain the exact asymptotic forms of them as t → ∞ that are valid uniformly in x > a for each order ν. If ν = ±1/2, we have well-known explicit expressions of them, which are often used in various circumstances, while otherwise there has been quite restricted information on them until quite recently. In the case when 0 ≤ x < a the distribution of σ a solves a boundary value problem of the associated second order differential equation on the finite interval (0, a) and the distribution of σ a or its density is represented by means of eigenfunction expansion ([3], [8], [12], etc.) and thereby we can obtain accurate estimates of them. In the case x > a, however, the region for the differential equation is the infinite interval (a, ∞) and the corresponding representation is given by a Fourier-Bessel transform (cf.[16]: Section 4.10), which it seems not simple a matter to derive an asymptotic form of the density directly from and there have been only a few partial results as given in [15], [18], [7] in which ν = 0 or/and relative ranges of x are restricted at least for sharp estimates (in addition to the cases ν = ±1/2). In the recent paper [2] Byczkowski, Malecki and Ryznar have computed an estimate of the density for σ a for all values of ν by using a certain integral representation of it given in [1]: they obtain upper and lower bounds of the correct order of magnitude valid uniformly for all t > 0, x > a, which however does not give the exact asymptotic form as we shall obtain in this paper (although in some cases their results are very close to and even finer than ours, see (i) of Remark 1 of the present paper). Hamana and Matumoto [10] have derived a similar (but, in a significant point, quite different) integral representation of the density of σ a for the case x > a (as 1 key words: first passage time, exterior problem, uniform estimate, Bessel diffusion