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Planar diffusions with rank-based characteristics and perturbed Tanaka equations
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  • Published: 09 May 2012

Planar diffusions with rank-based characteristics and perturbed Tanaka equations

  • E. Robert Fernholz1,
  • Tomoyuki Ichiba2,
  • Ioannis Karatzas1,3 &
  • …
  • Vilmos Prokaj4,5 

Probability Theory and Related Fields volume 156, pages 343–374 (2013)Cite this article

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  • 23 Citations

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Abstract

For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1(·), X 2(·)) with values in the plane and infinitesimal generator

$${\begin{array}{ll}\fancyscript{L}=\mathbf{1}_{\{ x_1 > x_2\}}\left(\frac{\rho^2}2{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_2^2}}}-h\frac{\partial}{\partial x_1} +g \frac{\partial}{\partial{x_2}}\right)\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathbf{1}_{\{ x_1\le x_2\}}\left(\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\rho^2}{2} {\frac{\partial^2}{\partial x{_2^2}}}+g\frac{\partial}{\partial x_1} - h \frac{\partial}{\partial{x_2}}\right),\,\,\,\,\,\,\,\,\,\,\,\, (0.1)\end{array}}$$

and discuss its realization in terms of appropriate systems of stochastic differential equations. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation

$$\begin{array}{ll}{\rm d}Z(t) = f \big(Z (t)\big){\rm d}M( t) + {\rm d}N(t), \quad Z(0) = \xi\end{array}$$

driven by suitable continuous, orthogonal semimartingales M(·) and N(·) and with f(·) of bounded variation, which we study here in detail; and those of a one-dimensional diffusion Y(·) with bang-bang drift \({dY(t) = -\lambda {\rm sign} \big( Y (t) \big) {\rm d}t + {\rm d}W (t), Y(0)=y}\) driven by a standard Brownian motion W(·). We also show that the planar diffusion (X 1(·), X 2(·)) can be represented in terms of this process Y(·), its local time L Y (·) at the origin, and an independent standard Brownian motion Q(·), in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.

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References

  1. Banner A.D., Fernholz E.R., Karatzas I.: Atlas models of equity markets. Ann. Appl. Probab. 15(4), 2296–2330 (2005). doi:10.1214/105051605000000449

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlow M.T.: One-dimensional stochastic differential equations with no strong solution. J. Lond. Math. Soc. (2) 26(2), 335–347 (1982). doi:10.1112/jlms/s2-26.2.335

    Article  MathSciNet  MATH  Google Scholar 

  3. Barlow M.T.: Skew Brownian motion and a one-dimensional stochastic differential equation. Stochastics 25(1), 1–2 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bass R.F., Pardoux É.: Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Relat. Fields 76(4), 557–572 (1987). doi:10.1007/BF00960074

    Article  MathSciNet  MATH  Google Scholar 

  5. Brossard, J., Leuridan, C.: Transformations browniennes et compléments indépendants: résultats et problèmes ouverts. In: Séminaire de Probabilités XLI. Lecture Notes in Math., vol. 1934, pp. 265–278. Springer, Berlin (2008). doi:10.1007/978-3-540-77913-1_13

  6. Chernyĭ A.S.: On strong and weak uniqueness for stochastic differential equations. Teor. Veroyatnost. i Primenen 46(3), 483–497 (2001). doi:10.1137/S0040585X97979093

    Article  MathSciNet  Google Scholar 

  7. Engelbert H.J.: On the theorem of T. Yamada and S. Watanabe. Stoch. Rep. 36(3–4), 205–216 (1991). doi:10.1080/17442509108833718

    MathSciNet  MATH  Google Scholar 

  8. Fernholz E.R.: Stochastic Portfolio Theory. Applications of Mathematics, vol. 48. Springer, New York (2002)

    Book  Google Scholar 

  9. Fernholz, E.R., Ichiba, T., Karatzas, I.: A second-order stock market model. Ann. Finance (2012, to appear). doi:10.1007/s10436-012-0193-2

  10. Fernholz, E.R., Ichiba, T., Karatzas. I., Prokaj, V.: Planar diffusions with rank-based characteristics: transition probabilities, time reversal, maximality and perturbed Tanaka equations. arXiv:1108.3992 (2011)

  11. Harrison J.M., Shepp L.A.: On skew Brownian motion. Ann. Probab. 9(2), 309–313 (1981). doi:10.1214/aop/1176994472

    Article  MathSciNet  MATH  Google Scholar 

  12. Ichiba, T., Karatzas, I., Shkolnikov, M.: Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Relat. Fields (2011, to appear). arXiv:1109.3823

  13. Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I., Fernholz, E.R.: Hybrid Atlas models. Ann. Appl. Probab. 21(2), 609–644 (2011). arXiv:0909.0065. doi:10.1214/10-AAP706

    Google Scholar 

  14. Karatzas I., Shreve S.E.: Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Probab. 12(3), 819–828 (1984). doi:10.1214/aop/1176993230

    Article  MathSciNet  MATH  Google Scholar 

  15. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)

  16. Krylov N.V.: On quasi diffusion processes. Theory Probab. Appl. 11, 373–389 (1966)

    Article  Google Scholar 

  17. Krylov, N.V.: Diffusion in the plane with reflection: construction of the process. Sibirski Mat. Zh. 10(2), 343–354 (in Russian). English translation in Sib. Math. J. 10(2), 244–252 (1969)

    Google Scholar 

  18. Krylov, N.V.: Itô’s stochastic integral equations. (Russian. English summary) Teor. Verojatnost. i Primenen 14, 340–348. English translation in Theor. Probab. Appl. 14, 330–336 (1969)

  19. Krylov N.V.: On weak uniqueness for some diffusions with discontinuous coefficients. Stoch. Process. Appl. 113(1), 37–64 (2004). doi:10.1016/j.spa.2004.03.012

    Article  MATH  Google Scholar 

  20. Le Gall, J.F.: Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In: Séminaire de Probabilités, XVII. Lecture Notes in Math., vol. 986, pp. 15–31. Springer, Berlin (1983)

  21. Lejay A.: On the constructions of the skew Brownian motion. Probab. Surv. 3, 413–466 (2006). doi:10.1214/154957807000000013

    Article  MathSciNet  MATH  Google Scholar 

  22. McKean H.P. Jr: Stochastic Integrals. Probability and Mathematical Statistics, vol. 5. Academic Press, New York (1969)

    Google Scholar 

  23. Nakao S.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9, 513–518 (1972)

    MathSciNet  MATH  Google Scholar 

  24. Pal, S., Pitman, J.: One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18(6), 2179–2207 (2008). arXiv:0704.0957. doi:10.1214/08-AAP516

    Google Scholar 

  25. Perkins, E.: Local time and pathwise uniqueness for stochastic differential equations. In: Seminar on Probability, XVI. Lecture Notes in Math., vol. 920, pp. 201–208. Springer, Berlin (1982)

  26. Prokaj, V.: The solution of the perturbed Tanaka-equation is pathwise unique. arXiv:1104.0740 (2011)

  27. Stroock D.W., Varadhan S.R.S.: Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften, vol. 233. Springer, Berlin (1979)

    Google Scholar 

  28. Veretennikov, A.Y.: Strong solutions of stochastic differential equations. Teor. Veroyatnost. i Primenen. 24(2), 348–360; translation in Theory Probab. Appl. 24, 354–366 (1979)

    Google Scholar 

  29. Veretennikov, A.Y.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 111(153)(3), 434–452; translation in Math. USSR-Sb. 39(3), 387–403 (1981). doi:10.1070/SM1981v039n03ABEH001522

  30. Veretennikov, A.Y.: Criteria for the existence of a strong solution of a stochastic equation. Teor. Veroyatnost. i Primenen. 27(3), 417–424; translation in Theory Probab. Appl. 27, 441–449 (1982)

  31. Walsh J.B.: A diffusion with a discontinuous local time. Temps Locaux. Astérisque 52(53), 37–45 (1978)

    Google Scholar 

  32. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. (N.S.) 93(135), 129–149; translation in Math. USSR Sb. 22, 129–149 (1974). doi:10.1070/SM1974v022n01ABEH001689

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Authors and Affiliations

  1. INTECH Investment Management LLC, One Palmer Square, Suite 441, Princeton, NJ, 08542, USA

    E. Robert Fernholz & Ioannis Karatzas

  2. Department of Statistics and Applied Probability, University of California, South Hall, Santa Barbara, CA, 93106, USA

    Tomoyuki Ichiba

  3. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Ioannis Karatzas

  4. Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Vilmos Prokaj

  5. Computer and Automation Institute of the Hungarian Academy of Sciences, Kende utca 13-17, 1111, Budapest, Hungary

    Vilmos Prokaj

Authors
  1. E. Robert Fernholz
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  4. Vilmos Prokaj
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Corresponding author

Correspondence to Vilmos Prokaj.

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Fernholz, E.R., Ichiba, T., Karatzas, I. et al. Planar diffusions with rank-based characteristics and perturbed Tanaka equations. Probab. Theory Relat. Fields 156, 343–374 (2013). https://doi.org/10.1007/s00440-012-0430-7

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  • Received: 01 September 2011

  • Accepted: 17 April 2012

  • Published: 09 May 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0430-7

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Keywords

  • Diffusion
  • Local time
  • Bang-bang drift
  • Lévy characterization of Brownian motion
  • Tanaka formulae
  • Weak and strong solutions
  • Skew representation
  • Skew Brownian motion
  • Modified and perturbed Tanaka equations

Mathematics Subject Classification

  • Primary 60H10
  • 60G44
  • Secondary 60J55
  • 60J60
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