Another Look at the Hartman-Watson Distributions

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Abstract

The article deals with the Hartman-Watson distributions and presents a new approach to them by investigating a special function u. The function u is strictly related to the distribution of the exponential functional of Brownian motion appearing in the mathematical finance framework. The study of the latter leads to new explicit representations for the function u. One of them is through a new parabolic PDE. Integral relations of convolution type between Hartman-Watson distributions and modified Bessel functions are presented. It turns out that u can be represented as an integral convolution of itself and the modified Bessel function K0. Finally, excursion theory and a subordinator connected to the hyperbolic cosine of Brownian motion are involved in order to obtain yet another representation for u. Possible applications of the resulting explicit formulas are discussed, among others Monte Carlo evaluations of u.

References

  1. 1.

    Alili L., Gruet J.-C.: An explanation of a generalized Bougerol’s identity in terms of hyperbolic Brownian motion. In: Yor, M. (ed.) Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers. Biblioteca de la Revista Matemática Iberoamericana (1997)

  2. 2.

    Barrieu, P., Rouault, A., Yor, M.: A study of the Hartman-Watson distribution motivated by numerical problems related to the pricing of Asian options. J. Appl. Probab. 41, 1049–1058 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bateman, H., Erdélyi, A.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954)

    Google Scholar 

  4. 4.

    Bernhart, G., Mai, J.: A note on the numerical evaluation of the Hartman-Watson density and distribution function. Innov. Quant. Risk Manag. 99, 337–345 (2015)

    Google Scholar 

  5. 5.

    Bertoin, J., Yor, M.: Exponential functionals of lévy processes. Probab. Surv. 2, 191–212 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Borodin, A., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae, 2nd ed. Birkhäuser (2002)

  7. 7.

    Donati-Martin, C., Ghomrasni, R., Yor, M.: On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options. Rev. Mat. Iberoam. 17, 179–193 (2001)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Eqworld - The World of Mathematical equations. http://eqworld.ipmnet.ru/index.htm

  9. 9.

    Geman H., Yor M.: Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375 (1993)

    Article  Google Scholar 

  10. 10.

    Gerhold, S.: The Hartman-Watson distribution revisited: asymptotics for pricing Asian options. J. Appl. Probab. 48, 892–899 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hartman, P.: Completely monotone families of solutions of nth order linear differential equations and infinitely divisible distributions. Ann. Scuola Norm. Sup. Pisa 4(3), 267–287 (1976)

    MATH  Google Scholar 

  12. 12.

    Hartman, P., Watson, G.: Normal distribution functions on spheres and the modified Bessel functions. Ann. Probab. 5, 582–585 (1974)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Integral calculator. http://www.integral-calculator.com

  14. 14.

    Kent, J.: The spectral decomposition of a diffusion hitting time. Ann. Probab. 10, 207–219 (1982)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Laforgia, A., Natalini, P.: Some inequalities for modified Bessel functions. J. Inequal. Appl. art 253035, pp. 10 (2010)

  16. 16.

    Lew, J.: On linear volterra integral equations of convolution type. Proc. Amer. Maths. Soc. 35, 450–455 (1972)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lyasoff, A.: Another look at the integral of exponential Brownian motion and the pricing of Asian options. Financ. Stochast. 20, 1061–1096 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mansuy, R., Yor, M.: Aspects of Brownian Motion. Springer, Universitext (2008)

    Google Scholar 

  19. 19.

    Matsumoto, H., Yor, M.: A Relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration. Osaka J. Math. 38, 383–398 (2001)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Matsumoto, H., Yor, M.: An analogue of Pitman’s 2M - X theorem for exponential Wiener functionals, part II: The role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86 (2001)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, I, Probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, II, Some related diffusion processes. Probab. Surv. 2, 348–384 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.11 of 2016-06-08

  25. 25.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2005)

    Google Scholar 

  26. 26.

    Schilling, R., Song, R., Vondraček, Z.: Bernstein Functions: Theory and Applications. de Gruyter (2010)

  27. 27.

    Wystup, U.: FX Options and structured products. Wiley, New York (2007)

  28. 28.

    Yor, M. (ed.): Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers, Biblioteca de la Revista Matemática Iberoamericana (1997)

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Acknowledgments

We would like to thank the anonymous referee for his/her careful reading of the manuscript and valuable remarks.

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Correspondence to Maciej Wiśniewolski.

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Jakubowski, J., Wiśniewolski, M. Another Look at the Hartman-Watson Distributions. Potential Anal 53, 1269–1297 (2020). https://doi.org/10.1007/s11118-019-09806-7

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Keywords

  • Hartman-Watson distributions
  • Additive functional of Brownian motion
  • Asian options
  • PDE
  • Excursions of Brownian motion
  • Lev́y measure
  • Modified Bessel functions

Mathematics Subject Classification (2010)

  • 60G40
  • 60G17
  • 91G80