Skip to main content
SpringerLink
Log in

Some Comments

  • Chapter
Henry P. McKean Jr. Selecta

Part of the book series: Contemporary Mathematicians ((CM))

  • 867 Accesses

Abstract

McK [49, ‘A free boundary problem for the heat equation arising from a problem in mathematical economics’] is my one and only excursion into “mathematical finance”. It is an appendix to P. Samuelson [90, ‘Rational theory of warrant pricing’] in which the correct recipe for pricing an American put option is worked out.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The numbers \(w = \sqrt{-\lambda }\) are the “fundamental tones” of a drum-head spanning ∂ D, whence Kac’s title.

Bibliography

  1. A. Ambrosetti and G. Prodi. On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl., 93:231–246, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bateman, H. Erdélyi. Higher Transcendental Functions, volume I. McGraw-Hill, New York/London, 1953.

    Google Scholar 

  3. M. Berger and P. Church. Complete integrability and perturbation of a nonlinear Dirichlet problem. Indiana Univ. Math. J., 28:935–952, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Black and M. Scholes. The pricing of options and corporate liabilities. J. Political Econ., 81:637–654, 1973.

    Article  MATH  Google Scholar 

  5. R. M. Blumenthal, R. K. Getoor, and H. P. McKean. Markov processes with identical hitting distributions. Illinois J. Math., 6:402–420, 1962.

    MathSciNet  MATH  Google Scholar 

  6. M. Bramson. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math., 31:531–581, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Buser, J. Conway, P. Doyle, and K. Semmler. Some planar isospectral domains. Internat. Math. Res. Notices, 9:301–400, 1994.

    MathSciNet  Google Scholar 

  8. R. Camassa and D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71(11):1661–1664, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. L. Chung. Probabilistic approach in potential theory to the equilibrium problem. Ann. Inst. Fourier, Grenoble, 23:313–322, 1973.

    Google Scholar 

  10. A. Constantin and H. P. McKean. A shallow water equation on the circle. Comm. Pure Appl. Math., 52:949–982, 1999.

    Article  MathSciNet  Google Scholar 

  11. J. Doob. Semi-martingales and subharmonic functions. Trans. AMS, 77:86–121, 1954.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Dym and H. P. McKean. Applications of the de Branges spaces of integral functions to the prediction of stationary Gaussian processes. Illinois J. Math., 14:299–343, 1970.

    MathSciNet  MATH  Google Scholar 

  13. H. Dym and H. P. McKean. Extrapolation and interpolation of stationary Gaussian processes. Ann. Math. Stat., 41:1819–1844, 1970.

    MathSciNet  Google Scholar 

  14. H. Dym and H. P. McKean. Gaussian Processes. Academic Press, 1976.

    MATH  Google Scholar 

  15. F. J. Dyson. Fredholm determinants and inverse scattering. Comm. Math. Phys., 47:171–183, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Ercolani and H. P. McKean. Geometry of KdV (4). Abel sums, Jacobi variety, and theta function in the scattering case. Invent. Math., 99(3):483–544, 1990.

    Google Scholar 

  17. J. Feldman, H. Knörrer, and E. Trubowitz. Riemann surfaces of infinite genus, volume 2 of CRM. AMS, Providence, 2002.

    Google Scholar 

  18. W. Feller. Generalized second order differential operators and their lateral conditions. Illinois J. Math., 1:459–564, 1957.

    MathSciNet  MATH  Google Scholar 

  19. R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:353–369, 1937.

    Google Scholar 

  20. I. M. Gelfand. Automorphic functions and the Theory of Representations. In Proc. Inter. Congress Math., volume 1, pages 74–85. Institut Mittag-Leffler, 1962.

    Google Scholar 

  21. P. Gilkey. Curvature and eigenvalues of the Laplacian for elliptic operators. Adv. Math., 10:344–382, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. P. Golchan. On some problems concerning Brownian motion in Lèvy’s sense. Teor. Veroyatnost. i Primen, 12:682–690, 1967.

    Google Scholar 

  23. C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via Riemannian orbofolds. Invent. Math., 110:1–22, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Hochstadt. Function-theoretic properties of the discriminant of Hill’s equation. Math. Zeit., 82:237–242, 1963.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Hochstadt. On the determination of Hill’s equation from its spectrum. Arch. Rat. Mech. Anal., 19:353–362, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Hunt. Markoff processes and potentials, I. Illinois J. Math., 1:44–93, 1957.

    MathSciNet  MATH  Google Scholar 

  27. G. Hunt. Markoff processes and potentials, III. Illinois J. Math., 2:151–213, 1958.

    MathSciNet  Google Scholar 

  28. K. Itô and H. P. McKean. Potentials and the random walk. Ill. J. Math., 4:119–132, 1960.

    MATH  Google Scholar 

  29. K. Itô and H. P. McKean. Brownian motion on a half line. Ill. J. Math., 7:181–231, 1963.

    MATH  Google Scholar 

  30. A. R. Its and V. B. Matveev. Hill operators with a finite number of lacunae. Funkcional. Anal. i Priložen., 9(1):69–70, 1975.

    Article  MathSciNet  Google Scholar 

  31. M. Kac. Probability theory: its role and impact. SIAM Review, 4:1–11, 1962.

    Article  MathSciNet  Google Scholar 

  32. M. Kac. Can you hear the shape of a drum? Amer. Math. Monthly, 73:1–23, 1966.

    Article  MATH  Google Scholar 

  33. M. Kac and D. Slepian. Large excursions of a Gaussian process. Ann. Math. Stat., 30:1215–1228, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  34. F. B. Knight. Random walks and a sojourn density process for Brownian motion. Trans. AMS, 167:56–86, 1963.

    Article  Google Scholar 

  35. A. Kolmogorov, I. Petrovskii, and N. Piscounov. Etude de l’èquation de la diffusion avec croissance de la quantitè de la matière. Bull. Math. Soc. Moscou, 1:1–35, 1937.

    Google Scholar 

  36. S. P. Lally and T. Selke. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Prob., 15:1052–1061, 1987.

    Article  Google Scholar 

  37. P. Lax. Integrals of non-linear equations and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Lax. Periodic solutions of the Korteweg-de Vries equation. Comm. Pure Appl. Math., 28:141–188, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  39. N. Levinson. Selected Papers of Norman Levinson, volume 2. Birkhauser-Verlag, Boston, 1998.

    Google Scholar 

  40. N. Levinson and H. P. McKean. Weighted trigonometrical approximation on \(\mathbb{R}^{1}\) with application to the germ field of a stationary gaussian noise. Acta Math., 112:99–143, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Lévy. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris, 1948.

    MATH  Google Scholar 

  42. P. Lévy. Random functions: general theory with special reference to Laplacian random functions. Univ. Cal. Pub. Stat., 1:331–390, 1953.

    Google Scholar 

  43. L. C. Li. Factorization problems on the Hilbert-Schmidt group and the Camassa-Holm equation. Comm. Pure Appl. Math., 56:186–209, 2008.

    Article  Google Scholar 

  44. R. R. London, H. P. McKean, L. C. G. Rogers, and D. Williams. A martingale approach to some Wiener-Hopf problems. Sem. Prob., Lecture Notes in Math., 920:41–90, 1982.

    Google Scholar 

  45. T. J. Lyons and H. P. McKean. Winding of the plane Brownian motion. Adv. Math., 51:212–225, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Malliavin. Stochastic Analysis, volume 313 of Grundlehren der Math. Wiss. Springer-Verlag, 1997.

    Google Scholar 

  47. H. P. McKean. A Hölder condition for Brownian local time. J. Math. Kyôto Univ., 1:195–201, 1962.

    MathSciNet  MATH  Google Scholar 

  48. H. P. McKean. A winding problem for a resonator driven by a white noise. J. Math. Kyôto Univ., 2:228–295, 1963.

    MathSciNet  Google Scholar 

  49. H. P. McKean. A free boundary problem for the heat equation arising from a problem in mathematical economics. Ind. Management Rev., 6:32–39, 1965.

    MathSciNet  Google Scholar 

  50. H. P. McKean. A probabilistic interpretation of equilibrium charge distributions. J. Math. Kyôto Univ., 4:617–625, 1965.

    MathSciNet  MATH  Google Scholar 

  51. H. P. McKean. A class of Markov processes associated with nonlinear parabolic equations. PNAS, 56:1907–1911, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  52. H. P. McKean. An upper bound of the spectrum of Δ on a manifold of negative curvature. J. Diff. Geom., 4:359–366, 1970.

    MathSciNet  MATH  Google Scholar 

  53. H. P. McKean. Selberg’s trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math., 25:225–246, 1972.

    Article  MathSciNet  Google Scholar 

  54. H. P. McKean. Geometry of differential space. Ann. Prob., 1:197–206, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  55. H. P. McKean. Wiener’s theory of nonlinear noise. SIAM-AMS Symp. Appl. Math., 6:191–209, 1973.

    MathSciNet  Google Scholar 

  56. H. P. McKean. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piscounov. Comm. Pure Appl. Math., 28:323–331, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  57. H. P. McKean. Brownian local times. Adv. Math., 16:91–111, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  58. H. P. McKean. The central limit theorem for Carleman’s equation. Israel J. Math., 21:54–92, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  59. H. P. McKean. An exponential formula for solving Bolzmann’s equation for a Maxwellian gas. J. Comb. Theory., 2:358–382, 1975.

    Article  MathSciNet  Google Scholar 

  60. H. P. McKean. Fluctuations in the kinetic theory of gases. Comm. Pure Appl. Math., 28:435–455, 1975.

    Article  MathSciNet  Google Scholar 

  61. H. P. McKean. Curvature of an \(\infty \)-dimensional manifold related to Hill’s equation. Jour. Diff. Geom., 17:523–529, 1982.

    MathSciNet  MATH  Google Scholar 

  62. H. P. McKean. Geometry of KdV (1): addition and the unimodal spectral class. Rev. Math. Iberoamer., 2:235–261, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  63. H. P. McKean. Geometry of KdV (2): three examples. J. Stat. Phys., 46:1115–1143, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  64. H. P. McKean. Curvatura integra, handle number, and genus of transcendental curves. Comm. Pure Appl. Math., 44:1057–1066, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  65. H. P. McKean. Geometry of KdV (3): Determinants and Unimodular Isospectral Flows. Comm. Pure Appl. Math., 45:389–415, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  66. H. P. McKean. How real is resonance? Comm. Pure Appl. Math., 50:317–322, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  67. H. P. McKean. Breakdown of a shallow water equation. Asian J. Math., 2:867–874, 1998.

    MathSciNet  MATH  Google Scholar 

  68. H. P. McKean. Correction to “Breakdown of a shallow water equation”. Asian J. Math., 3:3, 1999.

    Google Scholar 

  69. H. P. McKean. A quick proof of Riemann’s mapping theorem. Comm. Pure Appl. Math., 52:405–409, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  70. H. P. McKean. Brownian motion and the general diffusion: scale and clock. Bachelier Congress, pages 15–84, 2000.

    Google Scholar 

  71. H. P. McKean. Turbulence without pressure: existence of the invariant measure. Math. Appl. Anal, 9:463–468, 2002.

    MathSciNet  MATH  Google Scholar 

  72. H. P. McKean. Fredholm determinants and the Camassa-Holm hierarchy. Comm. Pure Appl. Math., 56:638–680, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  73. H. P. McKean. The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Comm. Pure Appl. Math., 56:998–1015, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  74. H. P. McKean. Book reviews: Riemann surfaces of infinite genus by J. Feldman, H. Knorrer and E. Trubowitz. Bull. Amer. Math. Soc., 42:79–87, 2005.

    Google Scholar 

  75. H. P. McKean and C. Scovel. Geometry of some simple non-linear differential operators. Ann. Scuola Norm. Sup. Pisa, 13:299–346, 1986.

    MathSciNet  MATH  Google Scholar 

  76. H. P. McKean and I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Diff. Geom., 1:43–69, 1967.

    MathSciNet  MATH  Google Scholar 

  77. H. P. McKean and E. Trubowitz. Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math., 29:143–226, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  78. H. P. McKean and E. Trubowitz. Hill surfaces and their theta functions. Bull. Amer. Math. Soc., 84:1042–1085, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  79. H. P. McKean and P. van Moerbeke. The spectrum of Hill’s equation. Inv. Math., 30:217–274, 1975.

    Article  MATH  Google Scholar 

  80. H. P. McKean and P. van Moerbeke. Hill and Toda curves. Comm. Pure Appl. Math., 33:23–42, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  81. F. G. Mehler. Über die Entwicklungen einer Funktion von beliebig vielen Variabeln nach Laplaceschen Funktionen höherer Ordnung. J. Reine Angew. Math., 66:161–176, 1866.

    Article  MathSciNet  MATH  Google Scholar 

  82. R. C. Merton. Theory of rational option pricing. Bell J. Econ. Managenent Sci., 4:141–183, 1973.

    Article  MathSciNet  Google Scholar 

  83. P. Myneni. The pricing of an american option. Ann. Appl. Prob., 2:1–23, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  84. S. P. Novikov. The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl., 8:236–246, 1974.

    Article  MATH  Google Scholar 

  85. V. K. Patodi. Curvature and the eigenforms of the Laplace operator. Jour. Diff. Geom., 5:233–249, 1971.

    MathSciNet  MATH  Google Scholar 

  86. L. Pitt and L. Tran. Local sample path properties of Gaussian fields. The Annals of Probability, 7:477–493, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  87. D. B. Ray. Sojourn times of a diffusion process. Illinois J. Math., 7:615–630, 1963.

    MathSciNet  MATH  Google Scholar 

  88. S. O. Rice. Mathematical analysis of random noise. Bell System Tech. J., 23:282, 1944.

    Article  MathSciNet  MATH  Google Scholar 

  89. S. O. Rice. Mathematical analysis of random noise. Bell System Tech. J., 24:46, 1945.

    Article  MathSciNet  MATH  Google Scholar 

  90. P. Samuelson. Rational theory of warrant pricing. Ind. Management Rev., 6:13–32, 1965.

    Google Scholar 

  91. P. Sarnak. Determinants of Laplacians, heights and finiteness. In P. Rabinowitz, editor, Analysis Etc., pages 601–622. Academic Press, 1990.

    Google Scholar 

  92. H. Schläfli. Ueber die allgemeine Möglichkeit der conformen Abbildung einer von Geraden begrenzten ebenen Figur in eine Halbebene. J. reine angew. Math., 78:63–80, 1874.

    MathSciNet  MATH  Google Scholar 

  93. R. Schoen, S. T. Yau, and S. Wolpert. Geometric bounds on the low eigenvalues of a compact manifold. Proc. Symp. Pure Math., 36:279–285, 1980.

    Article  MathSciNet  Google Scholar 

  94. A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc., 20:47–87, 1956.

    MathSciNet  MATH  Google Scholar 

  95. D. Sullivan and H. P. McKean. Brownian motion and harmonic functions on the class surface of the thrice-punctured sphere. Adv. Math., 51:203–211, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  96. H. Tanaka and H. P. McKean. Additive fractionals of the Brownian paths. Mem. Coll. Sci. Kyôto, 33:479–506, 1961.

    MathSciNet  MATH  Google Scholar 

  97. H. F. Trotter. A property of Brownian motion paths. Illinois J. Math., 2:425–433, 1958.

    MathSciNet  MATH  Google Scholar 

  98. S. Venakides. The infinite period limit of the inverse formalism for periodic potentials. Comm. Pure Appl. Math., 41:3–17, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  99. M. F. Vigneras. Variétés riemanniennes isospectrales et non isométriques. Ann. Math., 112:21–32, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  100. V. A. Volkonskii. Random time substitution in strong Markov processes. Teor. Veroyatnost. i Prim., 3:332–350, 1958.

    MathSciNet  Google Scholar 

  101. N. Wiener. Cybernetics. John Wiley & Sons, New York, 1948.

    Google Scholar 

  102. D. Williams. Markov properties of Brownian local time. Bull. Amer. Math. Soc., 75:1035–1036, 1964.

    Article  Google Scholar 

  103. D. Williams. Decomposition of the Brownian path. Bull. Amer. Math. Soc., 70:871–973, 1970.

    Article  Google Scholar 

  104. V. E. Zakharov. Collected Works. Dekker, New York & Basel, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute, New York University, New York, NY, USA

    Henry P. McKean Jr.

Authors
  1. Henry P. McKean Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henry P. McKean Jr. .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, California, USA

    F. Alberto Grünbaum

  2. Institut de Recherche en Math_ematique at Physique, Department of Mathematics, Universit_e catholique de Louvain, Louvain-la-Neuve, Belgium, & Brandeis University, Waltham, Massachusetts, USA

    Pierre van Moerbeke

  3. Tulane University, New Orleans, Louisiana, USA

    Victor H. Moll

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

McKean, H.P. (2015). Some Comments. In: Grünbaum, F., van Moerbeke, P., Moll, V. (eds) Henry P. McKean Jr. Selecta. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22237-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation