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
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
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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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
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
Barlow M.T.: Skew Brownian motion and a one-dimensional stochastic differential equation. Stochastics 25(1), 1–2 (1988)
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
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
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
Engelbert H.J.: On the theorem of T. Yamada and S. Watanabe. Stoch. Rep. 36(3–4), 205–216 (1991). doi:10.1080/17442509108833718
Fernholz E.R.: Stochastic Portfolio Theory. Applications of Mathematics, vol. 48. Springer, New York (2002)
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
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)
Harrison J.M., Shepp L.A.: On skew Brownian motion. Ann. Probab. 9(2), 309–313 (1981). doi:10.1214/aop/1176994472
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
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
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
Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)
Krylov N.V.: On quasi diffusion processes. Theory Probab. Appl. 11, 373–389 (1966)
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)
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)
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
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)
Lejay A.: On the constructions of the skew Brownian motion. Probab. Surv. 3, 413–466 (2006). doi:10.1214/154957807000000013
McKean H.P. Jr: Stochastic Integrals. Probability and Mathematical Statistics, vol. 5. Academic Press, New York (1969)
Nakao S.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9, 513–518 (1972)
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
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)
Prokaj, V.: The solution of the perturbed Tanaka-equation is pathwise unique. arXiv:1104.0740 (2011)
Stroock D.W., Varadhan S.R.S.: Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften, vol. 233. Springer, Berlin (1979)
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)
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
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)
Walsh J.B.: A diffusion with a discontinuous local time. Temps Locaux. Astérisque 52(53), 37–45 (1978)
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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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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DOI: https://doi.org/10.1007/s00440-012-0430-7
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