Small- t Expansion for the Hartman-Watson Distribution
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Small-t Expansion for the Hartman-Watson Distribution Dan Pirjol1 Received: 15 August 2019 / Revised: 15 August 2019 / Accepted: 28 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Hartman-Watson distribution with density fr (t) = I01(r) θ(r, t) with r > 0 is a probability distribution defined on t ∈ R+ , which appears in several problems of applied probability. The density of this distribution is given by an integral θ(r, t) which is difficult to evaluate numerically for small t → 0. Using saddle point methods, we obtain the first two terms of the t → 0 expansion of θ(ρ/t, t) at fixed ρ > 0. Keywords Asymptotic expansions · Saddle point method · Hartman-Watson distribution Mathematics Subject Classification (2010) 41A60 · 33F05 · 60-08
1 Introduction The Hartman-Watson distribution was introduced in the context of directional statistics (Hartman and Watson 1974) and was studied further in relation to the first hitting time of certain diffusion processes (Kent 1982). This distribution has received considerable attention due to its relation to the law of the time integral of the geometric Brownian motion, see Eq. (3) below (Yor 1992). The Hartman-Watson distribution is given in terms of the function θ(r, t) defined as θ(r, t) = √
r 2π 3 t
e
π2 2t
0
∞
ξ2
e− 2t e−r cosh ξ sinh ξ sin
πξ dξ . t
(1)
The normalized function fr (t) = θ(r,t) I0 (r) defines the density of a random variable taking values along the positive real axis t ≥ 0, distributed according to the Hartman-Watson law (Hartman and Watson 1974).
Dan Pirjol
[email protected] 1
School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA
Methodology and Computing in Applied Probability
The function θ(r, t) appears in the law of the additive functional of a standard Brownian motion Bt t (μ) At = e2(Bs +μs) ds . (2) 0
Functionals of this type appear in the pricing of Asian options in the Black-Scholes model (Dufresne 2000; 2005), in the study of diffusion processes in random media (Comtet et al. 1998), and in actuarial science (Boyle and Potapchik 2008). This integral appears also in the distributional properties of stochastic volatility models with log-normally distributed volatility, such as the β = 1 log-normal SABR model (Antonov et al. 2019) and the HullWhite model (Gulisashvili and Stein 2006, 2010). (μ) An explicit result for the joint distribution of (At , Bt ) was given by Yor (1992) 1 + e2x dudx (μ) μx− 12 μ2 t P(At ∈ du, Bt + μt ∈ dx) = e exp − θ(ex /u, t) , (3) 2u u where the function θ(r, t) is given by (1). A precise evaluation of θ(r, t) is required for the exact simulation of the time integral T 1 2 of the geometric Brownian motion AT = 0 eσ Wt +(r− 2 σ )t dt, conditional on the terminal value of the Brownian motion WT . This problem appears for example in the simulation of the β = 1 SABR model, see Cai et al. (2017). The paper (Cai et al. 2017) proposed an exact simulation method by inverting the Laplace transform of 1/AT . (μ) The Yor formula yields al
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