The Annals of Applied Probability

Analysis of market weights under volatility-stabilized market models

Soumik Pal

Full-text: Open access


We derive the joint density of market weights, at fixed times and suitable stopping times, of the volatility-stabilized market models introduced by Fernholz and Karatzas in [Ann. Finan. 1 (2005) 149–177]. The argument rests on computing the exit density of a collection of independent Bessel-square processes of possibly different dimensions from the unit simplex. We show that the law of the market weights is the same as that of the multi-allele Wright–Fisher diffusion model, well known in population genetics. Thus, as a side result, we furnish a novel proof of the transition density function of the Wright–Fisher model which was originally derived by Griffiths by bi-orthogonal series expansion.

Article information

Ann. Appl. Probab., Volume 21, Number 3 (2011), 1180-1213.

First available in Project Euclid: 2 June 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 91B28

Volatility-stabilized markets Bessel processes Wright–Fisher model Kelvin transform market weights


Pal, Soumik. Analysis of market weights under volatility-stabilized market models. Ann. Appl. Probab. 21 (2011), no. 3, 1180--1213. doi:10.1214/10-AAP725.

Export citation


  • [1] Axler, S., Bourdon, P. and Ramey, W. (2001). Harmonic Function Theory, 2nd ed. Graduate Texts in Mathematics 137. Springer, New York.
  • [2] Bass, R. F. and Perkins, E. A. (2003). Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc. 355 373–405 (electronic).
  • [3] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Probability and Its Applications. Birkhäuser, Basel.
  • [4] Bru, M.-F. (1991). Wishart processes. J. Theoret. Probab. 4 725–751.
  • [5] Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes (O. E. Barndorff-Nielson, T. Mikosch and S. I. Resnick, eds.) 41–55. Birkhäuser, Boston, MA.
  • [6] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123–159.
  • [7] Cox, J. C., Ingersoll Jr., J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–407.
  • [8] Durrett, R. (2008). Probability Models for DNA Sequence Evolution, 2nd ed. Probability and Its Applications (New York). Springer, New York.
  • [9] Eisenbaum, N. (1994). Dynkin’s isomorphism theorem and the Ray–Knight theorems. Probab. Theory Related Fields 99 321–335.
  • [10] Etheridge, A. M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI.
  • [11] Ethier, S. N. and Kurtz, T. G. (1981). The infinitely-many-neutral-alleles diffusion model. Adv. in Appl. Probab. 13 429–452.
  • [12] Ethier, S. N. and Kurtz, T. G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optim. 31 345–386.
  • [13] Fernholz, E. R. and Karatzas, I. K. (2005). Relative arbitrage in volatility-stabilized markets. Ann. Finan. 1 149–177.
  • [14] Fernholz, R. and Karatzas, I. (2009). Stochastic portfolio theory: An overview. In Handbook of Numerical Analysis (P. G. Ciarlet, ed.) XV. Special Volume: Mathematical Modelling and Numerical Methods in Finance (A. Bensoussan Q. Zhang, Guest eds.) 89–168. Elsevier, Amsterdam.
  • [15] Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3 349–375.
  • [16] Goia, I. (2009). Bessel and volatility-stabilized processes. Ph.D. thesis, Columbia Univ.
  • [17] Griffiths, R. C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 140–158.
  • [18] Griffiths, R. C. (1979). A transition density expansion for a multi-allele diffusion model. Adv. in Appl. Probab. 11 310–325.
  • [19] Hashemi, F. (2000). An evolutionary model of the size distribution of firms. J. Evol. Econ. 10 507–521.
  • [20] Ijiri, Y. and Simon, H. (1977). Interpretations of departures from the Pareto curve firm-size distributions. Journal of Political Economy 82 315–331.
  • [21] Jovanovic, B. (1982). Selection and the evolution of industry. Econometrica 50 649–670.
  • [22] König, W. and O’Connell, N. (2001). Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Comm. Probab. 6 107–114 (electronic).
  • [23] Pitman, J. (1996). Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299–329.
  • [24] Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425–457.
  • [25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [26] Shiga, T. and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 37–46.
  • [27] Simon, H. and Bonini, C. (1955). The size distribution of business firms. Amer. Econ. Rev. 48 607–617.
  • [28] Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 119–164.
  • [29] Warren, J. and Yor, M. (1998). The Brownian burglar: Conditioning Brownian motion by its local time process. In Séminaire de Probabilités, XXXII. Lecture Notes in Math. 1686 328–342. Springer, Berlin.
  • [30] Werner, W. (1995). Some remarks on perturbed reflecting Brownian motion. In Séminaire de Probabilités, XXIX. Lecture Notes in Math. 1613 37–43. Springer, Berlin.