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On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes

Authors: Paul M. N. Feehan and Camelia A. Pop
Journal: Trans. Amer. Math. Soc. 367 (2015), 7565-7593
MSC (2010): Primary 60G44, 60J60; Secondary 35K65
Published electronically: June 16, 2015
MathSciNet review: 3391893
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Abstract: Using results from a companion article by the authors (J. Differential Equations 254 (2013), 4401-4445) on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that the martingale problem associated with a degenerate-elliptic differential operator with unbounded, locally Hölder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Second, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable Hölder continuity properties. Third, for an Itô process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Itô process.

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  • [1] S. Altay and U. Schmock, Lecture notes on the Yamada-Watanabe condition for the pathwise uniqueness of solutions of certain stochastic differential equations, available at
  • [2] Alexandre Antonov, Timur Misirpashaev, and Vladimir Piterbarg, Markovian projection on a Heston model, J. Comput. Finance 13 (2009), no. 1, 23-47. MR 2557531 (2011c:60214)
  • [3] S. R. Athreya, M. T. Barlow, R. F. Bass, and E. A. Perkins, Degenerate stochastic differential equations and super-Markov chains, Probab. Theory Related Fields 123 (2002), 484-520.
  • [4] M. Atlan, Localizing volatilities, (2006), arXiv:math/0604316.
  • [5] Richard F. Bass, Krzysztof Burdzy, and Zhen-Qing Chen, Pathwise uniqueness for a degenerate stochastic differential equation, Ann. Probab. 35 (2007), no. 6, 2385-2418. MR 2353392 (2009c:60144),
  • [6] Richard F. Bass and Alexander Lavrentiev, The submartingale problem for a class of degenerate elliptic operators, Probab. Theory Related Fields 139 (2007), no. 3-4, 415-449. MR 2322703 (2008f:60060),
  • [7] Richard F. Bass and Edwin A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc. 355 (2003), no. 1, 373-405 (electronic). MR 1928092 (2003m:60144),
  • [8] R. F. Bass and E. A. Perkins, Countable systems of degenerate stochastic differential equations with applications to super-Markov chains, Electron. J. Probab. 9 (2004), 634-673.
  • [9] Erhan Bayraktar, Constantinos Kardaras, and Hao Xing, Valuation equations for stochastic volatility models, SIAM J. Financial Math. 3 (2012), no. 1, 351-373. MR 2968038,
  • [10] A. Bentata and R. Cont, Mimicking the marginal distributions of a semimartingale, arXiv:0910.3992.
  • [11] V. I. Bogachev, N. V. Krylov, and M. Röckner, Elliptic and parabolic equations for measures, Uspekhi Mat. Nauk 64 (2009), no. 6(390), 5-116 (Russian, with Russian summary); English transl., Russian Math. Surveys 64 (2009), no. 6, 973-1078. MR 2640966 (2011c:35592),
  • [12] G. Brunick, A weak existence result with application to the financial engineer's calibration problem, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA, May 2008.
  • [13] Gerard Brunick, Uniqueness in law for a class of degenerate diffusions with continuous covariance, Probab. Theory Related Fields 155 (2013), no. 1-2, 265-302. MR 3010399,
  • [14] Gerard Brunick and Steven Shreve, Mimicking an Itô process by a solution of a stochastic differential equation, Ann. Appl. Probab. 23 (2013), no. 4, 1584-1628. MR 3098443
  • [15] Alexander S. Cherny and Hans-Jürgen Engelbert, Singular stochastic differential equations, Lecture Notes in Mathematics, vol. 1858, Springer-Verlag, Berlin, 2005. MR 2112227 (2005j:60002)
  • [16] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), no. 4, 899-965. MR 1623198 (99d:35182),
  • [17] Panagiota Daskalopoulos and Eunjai Rhee, Free-boundary regularity for generalized porous medium equations, Commun. Pure Appl. Anal. 2 (2003), no. 4, 481-494. MR 2019063 (2004i:35318),
  • [18] B. Dupire, Pricing with a smile, Risk Magazine 7 (1994), 18-20.
  • [19] H. J. Engelbert and W. Schmidt, On one-dimensional stochastic differential equations with generalized drift, Stochastic differential systems (Marseille-Luminy, 1984) Lecture Notes in Control and Inform. Sci., vol. 69, Springer, Berlin, 1985, pp. 143-155. MR 798317 (86m:60144),
  • [20] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986. MR 838085 (88a:60130)
  • [21] P. M. N. Feehan, Maximum principles for boundary-degenerate linear parabolic differential operators, arXiv:1306.5197.
  • [22] Paul M. N. Feehan, Maximum principles for boundary-degenerate second order linear elliptic differential operators, Comm. Partial Differential Equations 38 (2013), no. 11, 1863-1935. MR 3169765,
  • [23] P. M. N. Feehan, Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations, arXiv:1305.5098.
  • [24] P. M. N. Feehan and C. A. Pop, Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Itô processes, arXiv:1112.4824v1.
  • [25] P. M. N. Feehan and C. A. Pop, On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Itô processes, arXiv:1211.4636v1.
  • [26] Paul M. N. Feehan and Camelia A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients, J. Differential Equations 254 (2013), no. 12, 4401-4445. MR 3040945,
  • [27] A. Friedman, Stochastic differential equations and applications, vol. I, II, Academic, New York, 1975 and 1976.
  • [28] I. Gyöngy, Mimicking the one-dimensional marginal distributions of processes having an Itô differential, Probab. Theory Relat. Fields 71 (1986), no. 4, 501-516. MR 833267 (87k:60147),
  • [29] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6 (1993), 327-343.
  • [30] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam, 1981. MR 637061 (84b:60080)
  • [31] Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940 (92h:60127)
  • [32] H. Koch, Non-Euclidean singular integrals and the porous medium equation, Habilitation Thesis, University of Heidelberg, 1999,
  • [33] N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York, 1980. Translated from the Russian by A. B. Aries. MR 601776 (82a:60062)
  • [34] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, American Mathematical Society, Providence, RI, 1996.
  • [35] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations 35 (2010), no. 1, 1-22. MR 2748616 (2011m:35142),
  • [36] Dejun Luo, Pathwise uniqueness of multi-dimensional stochastic differential equations with Hölder diffusion coefficients, Front. Math. China 6 (2011), no. 1, 129-136. MR 2762976 (2011k:60205),
  • [37] Nikolai Nadirashvili, Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 3, 537-549. MR 1612401 (99b:35042)
  • [38] Bernt Øksendal, Stochastic differential equations: An introduction with applications, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. MR 2001996 (2004e:60102)
  • [39] V. Piterbarg, Markovian projection method for volatility calibration, Risk Magazine (April 2007), 84-89,
  • [40] Philip E. Protter, Stochastic integration and differential equations, Second edition, Stochastic Modelling and Applied Probability, vol. 21, Springer-Verlag, Berlin, 2005. Version 2.1; Corrected third printing. MR 2273672 (2008e:60001)
  • [41] Michael Röckner and Xicheng Zhang, Weak uniqueness of Fokker-Planck equations with degenerate and bounded coefficients, C. R. Math. Acad. Sci. Paris 348 (2010), no. 7-8, 435-438 (English, with English and French summaries). MR 2607035 (2011a:82080),
  • [42] M. Rutkowski, On solutions of stochastic differential equations with drift, Probab. Theory Related Fields 85 (1990), no. 3, 387-402. MR 1055762 (91e:60173),
  • [43] Ming Shi, Local intensity and its dynamics in multi-name credit derivatives modeling, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)-Rutgers, The State University of New Jersey - New Brunswick. MR 2736653
  • [44] Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin, 1979. MR 532498 (81f:60108)
  • [45] Jin Wang, Semimartingales, Markov processes and their applications in mathematical finance, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)-Rutgers, The State University of New Jersey - New Brunswick. MR 2827371
  • [46] Toshio Yamada, Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien, Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), Lecture Notes in Math., vol. 649, Springer, Berlin, 1978, pp. 114-131 (French). MR 520000 (80k:60076)

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Additional Information

Paul M. N. Feehan
Affiliation: Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

Camelia A. Pop
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395

Keywords: Degenerate-parabolic differential operator, degenerate diffusion process, Heston stochastic volatility process, degenerate martingale problem, mathematical finance, mimicking one-dimensional marginal probability distributions, degenerate stochastic differential equation
Received by editor(s): February 11, 2013
Received by editor(s) in revised form: June 25, 2013
Published electronically: June 16, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1059206. The second author was partially supported by a Rutgers University fellowship.
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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