On the one-sided Tanaka equation with drift

Ioannis Karatzas (Intech Investment Management, Columbia University)
Albert N. Shiryaev (Steklov Mathematical Institute)
Mykhaylo Shkolnikov (Intech Investment Management, Stanford University)

Abstract


We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift lambda. We observe a dichotomy in terms of the values of the drift parameter: for $\lambda\leq 0$, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for $\lambda > 0$, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable ``Brownian perturbations".

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Pages: 664-677

Publication Date: October 31, 2011

DOI: 10.1214/ECP.v16-1665

References

  1. Appuhamillage, T., Bokil, V., Thomann, E., Waymire, E. and Wood, B. (2011). Occupation and local times for skew Brownian motion with application to dispersion along an interface. Annals of Applied Probability 21 , 183-214. MR2759199
  2. Barlow, M.T. (1988). Skew Brownian motion and a one-dimensional stochastic differential equation. Stochastics 25 , 1-2. MR1008231
  3. Bass, R.F., Burdzy, K. and Chen, Z.Q. (2007). Pathwise uniqueness for a degenerate stochastic differential equation. Annals of Probability 35 , 2385-2418. MR2353392
  4. Bass, R. and Pardoux, E. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probability Theory and Related Fields 76 , 557-572. MR0917679
  5. Chitashvili, R. J. (1997). On the nonexistence of a strong solution in the boundary problem for a sticky Brownian motion. Proc. A. Razmadze Math. Inst. 115 , 17-31. (Available in preprint form as CWI Technical Report BS-R8901 (1989), Centre for Mathematics and Computer Science, Amsterdam.) MR1639096
  6. Engelbert, H.J. and Schmidt, W. (1981). On the behaviour of certain functionals of the Wiener process and applications to stochastic differential equations. Lecture Notes in Control and Information Sciences 36 , 47-55. Springer-Verlg, Berlin. MR0653645
  7. Engelbert, H.J. and Schmidt, W. (1984). On one-dimensional stochastic differential equations with generalized drift. Lecture Notes in Control and Information Sciences 69 , 143-155. Springer-Verlg, Berlin. MR0798317
  8. Engelbert, H.J. and Schmidt, W. (1985). On solutions of stochastic differential equations without drift. Zeitschrift fuer Wahrscheinlichkeitstheorie und verwandte Gebiete 68 , 287-317. MR0771468
  9. Ethier, S. N. and Kurtz, T. G. (1986). Markov processes: Characterization and Convergence. J. Wiley and Sons, New York. MR0838085
  10. Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468-519. MR0047886
  11. Fernholz, E.R., Ichiba, T., Karatzas, I. and Prokaj, V. (2011). Planar diffusions with rank-based characteristics, and perturbed Tanaka equations. Preprint, Intech Investment Management, Princeton. Available at http://arxiv.org/abs/1108.3992 .
  12. Gihman, I.I. and Skorohod, A.V. (1972). Stochastic Differential Equations. Springer Verlag, New York. MR0263172
  13. Girsanov, I.V. (1962). An example of non-uniqueness of the solution to the stochastic differential equation of K. Ito. Theory of Probability and Its Applications 7 , 325-331.
  14. Hajek, B. (1985). Mean stochastic comparison of diffusions. Zeitschrift fuer Wahrscheinlichkeitstheorie und verwandte Gebiete 68 , 315-329. MR0771469
  15. Harrison, J.M. and Shepp, L.A. (1981). On skew Brownian motion. Annals of Probability 9 , 309-313. MR0606993
  16. Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus. Second Edition, Springer Verlag, New York. MR1121940
  17. Lejay, A. (2006). On the constructions of the skew Brownian motion. Probability Surveys 3 , 413-466. MR2280299
  18. Mandl, P. (1968). Analytical treatment of One-Dimensional Markov Processes. Springer Verlag, New York. MR0247667
  19. Manabe, Sh. and Shiga, T. (1973). On one-dimensional stochastic differential equations with non-sticky boundary conditions. Journal of Mathematics Kyoto University 13 , 595-603. MR0346907
  20. Nakao, S. (1972). On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka Journal of Mathematics 9 , 513-518. MR0326840
  21. Prokaj, V. (2010). The solution of the perturbed Tanaka equation is pathwise unique. Annals of Probability , to appear.
  22. Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Third Edition, Springer Verlag, New York. MR1725357
  23. Walsh, J.B. (1978). A diffusion with a discontinuous local time. In Temps Locaux. Asterisque 52-53 , 37-45.
  24. Warren, J. (1997). Branching processes, the Ray-Knight theorem, and sticky Brownian motion. Lecture Notes in Mathematics 1655 , 1-15. MR1478711
  25. Warren, J. (1999). On the joining of sticky Brownian motion. Lecture Notes in Mathematics 1709 , 257-266. Springer-Verlg, Berlin. MR1767999
  26. Zvonkin, A.K. (1974). A transformation on the state-space of a diffusion process that removes the drift. Mathematics of the USSR (Sbornik) 22 , 129-149.
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