Abstract
This paper concerns the density of the Hartman–Watson law. Yor (Z Wahrsch Verw Gebiete 53:71–95, 1980) obtained an integral formula that gives a closed-form expression of the Hartman–Watson density. In this paper, based on Yor’s formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a result of Dufresne (Adv Appl Probab 33:223–241, 2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion.
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References
Alili, L., Gruet, J.-C.: An explanation of a generalized Bougerol’s identity in terms of hyperbolic Brownian motion. In: Exponential Functionals and Principal Values Related to Brownian Motion: A Collection of Research Papers, Yor, M. (ed.), pp. 15–33, Biblioteca de la Revista Matemática Iberoamericana, Rev. Mat. Iberoamericana, Madrid (1997)
Bernhart, G., Mai, J.-F.: A note on the numerical evaluation of the Hartman–Watson density and distribution function. In: Glau, K., Scherer, M., Zagst, R. (eds.) Innovations in Quantitative Risk Management, pp. 337–345. Springer, Cham (2015)
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, corrected reprint of 2nd ed., 2002, Birkhäuser, Basel (2015)
Carr, P., Schröder, M.: Bessel processes, the integral of geometric Brownian motion, and Asian options, Teor. Veroyatnost. i Primenen. 48, 503–533 (2003); translation in Theory of Probab. Appl. 48, 400–425 (2004)
Dufresne, D.: The integral of geometric Brownian motion. Adv. Appl. Probab. 33, 223–241 (2001)
Hariya, Y.: On some identities in law involving exponential functionals of Brownian motion and Cauchy random variable. Stoch. Process. Appl. 130, 5999–6037 (2020)
Hartman, P., Watson, G.S.: “Normal” distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2, 593–607 (1974)
Jakubowski, J., Wiśniewolski, M.: On hyperbolic Bessel processes and beyond. Bernoulli 19, 2437–2454 (2013)
Jakubowski, J., Wiśniewolski, M.: Another look at the Hartman–Watson distributions. Potential Anal. 53, 1269–1297 (2020)
Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972)
Lyasoff, A.: Another look at the integral of exponential Brownian motion and the pricing of Asian options. Finance Stoch. 20, 1061–1096 (2016)
Matsumoto, H., Yor, M.: An analogue of Pitman’s \(2M-X\) theorem for exponential Wiener functionals, Part I: a time-inversion approach. Nagoya Math. J. 159, 125–166 (2000)
Matsumoto, H., Yor, M.: On Dufresne’s relation between the probability laws of exponential functionals of Brownian motions with different drifts. Adv. Appl. Probab. 35, 184–206 (2003)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, I: probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, II: some related diffusion processes. Probab. Surv. 2, 348–384 (2005)
Schröder, M.: On the integral of geometric Brownian motion. Adv. Appl. Probab. 35, 159–183 (2003)
Vakeroudis, S.: Bougerol’s identity in law and extensions. Probab. Surv. 9, 411–437 (2012)
Yor, M.: Loi de l’indice du lacet Brownien, et distribution de Hartman–Watson. Z. Wahrsch. Verw. Gebiete 53, 71–95 (1980)
Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531 (1992), also in: [20], pp. 23–48
Yor, M.: Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin (2001)
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Appendix
Appendix
We complete the proof of Lemma 2.1.
Proof of (ii) \(\Rightarrow \) (iii) in Lemma 2.1 Given \(x\in \mathbb {R}\), we integrate both sides of (2.3) multiplied by \(e^{-r\cosh x}\) with respect to \(r\ge 0\). Then by Fubini’s theorem, the left-hand side turns into that of (2.4). Therefore, it suffices to show that
By the latter condition in (2.1), we may also use Fubini’s theorem to rewrite the left-hand side of the claimed identity (A.1) as
On the other hand, by the symmetry of \(G\), the right-hand side of (A.1) is equal to
Noting the fact that
for any \(x,y\in \mathbb {R}\), we compare the last expression with (A.2) to conclude identity (A.1). \(\square \)
We turn to the proof of the implication from (iii) to (i). To this end, we prepare the following lemma:
Lemma A.1
For every \(x, b\in \mathbb {R}\), it holds that
Proof
We may assume \(|x|\ne |b|\); validity in the case \(|x|=|b|\) is verified by passing to the limit. By symmetrization and by the relation \( \cosh (2b)+\cosh (2y)=2\left( \cosh ^{2}b+\sinh ^{2}y \right) \), the left-hand side of the claimed identity is equal to
which is rewritten, due to relation (A.3), as
where we changed the variables with \(\sinh y=z\) in the second line. Now the claimed identity follows. \(\square \)
We are prepared to finish the proof of Lemma 2.1.
Proof of (iii) \(\Rightarrow \) (i) in Lemma 2.1 We appeal to the injectivity of Fourier transform. For this purpose, we first observe that
Indeed, the former observation is immediate from (2.6) and the former condition in (2.1), while the latter is clear by the latter condition in (2.1). For an arbitrarily fixed \(\xi \in \mathbb {R}\), we integrate both sides of (2.4) multiplied by \(\cos (\xi x)\) with respect to \(x\in \mathbb {R}\). Then by the latter finiteness in (A.4) and Fubini’s theorem, the right-hand side turns into
where we used the fact that
which is verified by standard residue calculus. On the other hand, as for the left-hand side of (2.4), we have
where we used Lemma A.1 for the second line, Fubini’s theorem for the third thanks to the former finiteness in (A.4), and fact (A.6) for the fourth. Since the last expression agrees with (A.5) for any \(\xi \in \mathbb {R}\) and the function \(G\) is assumed to be symmetric, the injectivity of Fourier transform entails relation (2.2). The proof completes. \(\square \)
Remark A.1
By using Lemma A.1, implication (i) \(\Rightarrow \) (iii) may also be proven in the same manner as in the proof of (i) \(\Rightarrow \) (ii) given in Sect. 2.
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Hariya, Y. Integral Representations for the Hartman–Watson Density. J Theor Probab 35, 209–230 (2022). https://doi.org/10.1007/s10959-020-01067-0
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DOI: https://doi.org/10.1007/s10959-020-01067-0