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Forward invariance and Wong–Zakai approximation for stochastic moving boundary problems

  • Martin Keller-Ressel
  • Marvin S. MüllerEmail author
Article
  • 8 Downloads

Abstract

We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly nonlinear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong–Zakai-type approximation. After a coordinate transformation, the problems are reformulated and analyzed in terms of stochastic evolution equations on domains of fractional powers of linear operators.

Keywords

Stochastic partial differential equation Stefan problem moving boundary problem Phase separation Forward invariance Wong–Zakai approximation 

Mathematics Subject Classification

60H15 35R60 

Notes

References

  1. 1.
    H. Amann. Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J. Math. Anal. Appl., 65(2):432–467, 1978.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Appell and P. P. Zabrejko. Nonlinear superposition operators, volume 95 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990.Google Scholar
  3. 3.
    R. Cont and M. S. Müller. A stochastic PDE model for limit order book dynamics. arXiv preprint arXiv:1904.03058, 2019.
  4. 4.
    G. Da Prato, S. Kwapień, and J. Zabczyk. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics, 23(1):1–23, 1987.MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Da Prato and J. Zabczyk. A note on stochastic convolution. Stochastic Analysis and Applications, 10(2):143–153, 1992.MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1992.Google Scholar
  7. 7.
    K. Deimling. Ordinary differential equations in Banach spaces. Lecture Notes in Mathematics, Vol. 596. Springer-Verlag, Berlin-New York, 1977.Google Scholar
  8. 8.
    N. Dunford and J. T. Schwartz. Linear operators. Part II. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication.Google Scholar
  9. 9.
    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.Google Scholar
  10. 10.
    L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.Google Scholar
  11. 11.
    D. Filipović, S. Tappe, and J. Teichmann. Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financial Math., 1(1):523–554, 2010.MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Grisvard. Caractérisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal., 25:40–63, 1967.MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Hambly, J. Kalsi, and J. Newbury. Limit order books, diffusion approximations and reflected spdes: from microscopic to macroscopic models. arXiv preprint arXiv:1808.07107, 2018.
  14. 14.
    D. Henry. Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1981.Google Scholar
  15. 15.
    N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.Google Scholar
  16. 16.
    W. Jachimiak. A note on invariance for semilinear differential equations. Bull. Polish Acad. Sci. Math., 45(2):181–185, 1997.MathSciNetzbMATHGoogle Scholar
  17. 17.
    O. Kallenberg. Foundations of Modern Probability. Applied probability. Springer, 2002.CrossRefGoogle Scholar
  18. 18.
    M. Keller-Ressel and M. S. Müller. A Stefan-type stochastic moving boundary problem. Stochastics and Partial Differential Equations: Analysis and Computations, 4(4):746–790, 2016.MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Kunze and J. van Neerven. Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations. J. Differential Equations, 253(3):1036–1068, 2012.MathSciNetCrossRefGoogle Scholar
  20. 20.
    J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications - 1. Springer, Springer, 1972.CrossRefGoogle Scholar
  21. 21.
    A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Basel, 1995.CrossRefGoogle Scholar
  22. 22.
    A. Lunardi. Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, second edition, 2009.Google Scholar
  23. 23.
    A. Milian. Comparison theorems for stochastic evolution equations. Stoch. Stoch. Rep., 72(1-2):79–108, 2002.MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. S. Müller. Semilinear stochastic moving boundary problems. Doctoral thesis, TU Dresden, 2016.Google Scholar
  25. 25.
    M. S. Müller. A stochastic Stefan-type problem under first-order boundary conditions. The Annals of Applied Probability, 28(4):2335–2369, 2018.MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete and Continuous Dynamical Systems - B, 24(8):4317–4339, 2019.MathSciNetCrossRefGoogle Scholar
  27. 27.
    T. Nakayama. Support theorem for mild solutions of SDE’s in Hilbert spaces. J. Math. Sci. Univ. Tokyo, 11(3):245–311, 2004.MathSciNetzbMATHGoogle Scholar
  28. 28.
    T. Nakayama. Viability theorem for SPDE’s including HJM framework. J. Math. Sci. Univ. Tokyo, 11(3):313–324, 2004.MathSciNetzbMATHGoogle Scholar
  29. 29.
    N. Pavel. Invariant sets for a class of semi-linear equations of evolution. Nonlinear Anal., 1(2):187–196, 1976/77.MathSciNetCrossRefGoogle Scholar
  30. 30.
    N. Pavel. Differential equations, flow invariance and applications, volume 113. Pitman Pub., 1984.Google Scholar
  31. 31.
    A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Number 44 in Applied Mathematical Sciences. Springer, 1992.Google Scholar
  32. 32.
    J. Prüss. On semilinear parabolic evolution equations on closed sets. J. Math. Anal. Appl., 77(2):513–538, 1980.MathSciNetCrossRefGoogle Scholar
  33. 33.
    M. Sauer and W. Stannat. Analysis and approximation of stochastic nerve axon equations. Mathematics of Computation, 2016.Google Scholar
  34. 34.
    J. Stefan. Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Wien. Ber. XCVIII, Abt. 2a (965–983), 1888.Google Scholar
  35. 35.
    D. W. Stroock and S. R. S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 333–359. Univ. California Press, Berkeley, Calif., 1972.Google Scholar
  36. 36.
    G. Tessitore and J. Zabczyk. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ., 6(4):621–655, 2006.MathSciNetCrossRefGoogle Scholar
  37. 37.
    K. Twardowska. Wong-Zakai approximations for stochastic differential equations. Acta Appl. Math., 43(3):317–359, 1996.MathSciNetCrossRefGoogle Scholar
  38. 38.
    T. Valent. Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, volume 31 of Springer Tracts in Natural Philosophy. Springer New York, 1988.Google Scholar
  39. 39.
    T. Valent. A property of multiplication in Sobolev spaces. Some applications. Rend. Sem. Mat. Univ. Padova, 74:63–73, 1985.Google Scholar
  40. 40.
    J. M. A. M. van Neerven, M. C. Veraar, and L. Weis. Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal., 255(4):940–993, 2008.MathSciNetCrossRefGoogle Scholar
  41. 41.
    E. Wong and M. Zakai. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci., 3:213–229, 1965.MathSciNetCrossRefGoogle Scholar
  42. 42.
    J. Zabczyk. Stochastic invariance and consistency of financial models. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11(2):67–80, 2000.Google Scholar
  43. 43.
    Z. Zheng. STOCHASTIC STEFAN PROBLEMS: EXISTENCE, UNIQUENESS, AND MODELING OF MARKET LIMIT ORDERS. PhD thesis, Graduate College of the University of Illinois at Urbana-Champaign, 2012.Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical StochasticTU DresdenDresdenGermany
  2. 2.Department of MathematicsETH ZürichZurichSwitzerland
  3. 3.Zenai AGZurichSwitzerland

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