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August  2019, 24(8): 4317-4339. doi: 10.3934/dcdsb.2019121

Approximation of the interface condition for stochastic Stefan-type problems

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland

Dedicated to Professor Peter E. Kloeden on the occasion of his 70th birthday

Received  March 2018 Revised  October 2018 Published  June 2019

Fund Project: The author acknowledges support by the Swiss National Science Foundation through grant SNF 205121 163425.

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

Citation: Marvin S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4317-4339. doi: 10.3934/dcdsb.2019121
References:
[1]

R. ContA. Kukanov and S. Stoikov, The price impact of order book events, Journal of Financial Econometrics, 12 (2014), 47-88.   Google Scholar

[2]

G. da Prato and J. Zabczyk, A note on stochastic convolution, Stochastic Analysis and Applications, 10 (1992), 143-153.  doi: 10.1080/07362999208809260.  Google Scholar

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K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194, Springer-Verlag, New York, 2000.  Google Scholar

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B. Hambly and J. Kalsi, A reflected moving boundary problem driven by space-time white noise, arXiv: 1805.10166. Google Scholar

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B. Hambly and J. Kalsi, Stefan problems for reflected spdes driven by space-time white noise, arXiv: 1806.04739. Google Scholar

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[7]

M. Keller-Ressel and M. Müller, A Stefan-type stochastic moving boundary problem, Stochastics and Partial Differential Equations: Analysis and Computations, 4 (2016), 746-790.  doi: 10.1007/s40072-016-0076-z.  Google Scholar

[8]

M. Keller-Ressel and M. Müller, Forward-invariance and Wong-Zakai approximation for stochastic moving boundary problems, arXiv: 1801.05203. Google Scholar

[9]

K. KimZ. Zheng and R. Sowers, A stochastic Stefan problem, Journal of Theoretical Probability, 25 (2012), 1040-1080.  doi: 10.1007/s10959-011-0392-1.  Google Scholar

[10]

M. Kunze and J. van Neerven, Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations, Journal of Differential Equations, 253 (2012), 1036-1068.  doi: 10.1016/j.jde.2012.04.013.  Google Scholar

[11]

A. Lipton, U. Pesavento and M. Sotiropoulos, Trading strategies via book imbalance, Risk, 70. Google Scholar

[12]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16, Springer Science & Business Media, 1995.  Google Scholar

[13]

A. Lunardi, Interpolation Theory, , Edizioni della Normale, Pisa, 2009.  Google Scholar

[14]

M. Müller, A stochastic Stefan-type problem under first-order boundary conditions, The Annals of Applied Probability, 28 (2018), 2335-2369.  doi: 10.1214/17-AAP1359.  Google Scholar

[15]

J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Wien. Ber., 98 (1888), 965-983.   Google Scholar

[16]

J. Van NeervenM. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, Journal of Functional Analysis, 255 (2008), 940-993.  doi: 10.1016/j.jfa.2008.03.015.  Google Scholar

[17]

Z. Zheng, Stochastic Stefan Problems: Existence, Uniqueness, and Modeling of Market Limit Orders, Ph.D thesis, University of Illinois at Urbana-Champaign, 2012.  Google Scholar

show all references

References:
[1]

R. ContA. Kukanov and S. Stoikov, The price impact of order book events, Journal of Financial Econometrics, 12 (2014), 47-88.   Google Scholar

[2]

G. da Prato and J. Zabczyk, A note on stochastic convolution, Stochastic Analysis and Applications, 10 (1992), 143-153.  doi: 10.1080/07362999208809260.  Google Scholar

[3]

K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194, Springer-Verlag, New York, 2000.  Google Scholar

[4]

B. Hambly and J. Kalsi, A reflected moving boundary problem driven by space-time white noise, arXiv: 1805.10166. Google Scholar

[5]

B. Hambly and J. Kalsi, Stefan problems for reflected spdes driven by space-time white noise, arXiv: 1806.04739. Google Scholar

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[7]

M. Keller-Ressel and M. Müller, A Stefan-type stochastic moving boundary problem, Stochastics and Partial Differential Equations: Analysis and Computations, 4 (2016), 746-790.  doi: 10.1007/s40072-016-0076-z.  Google Scholar

[8]

M. Keller-Ressel and M. Müller, Forward-invariance and Wong-Zakai approximation for stochastic moving boundary problems, arXiv: 1801.05203. Google Scholar

[9]

K. KimZ. Zheng and R. Sowers, A stochastic Stefan problem, Journal of Theoretical Probability, 25 (2012), 1040-1080.  doi: 10.1007/s10959-011-0392-1.  Google Scholar

[10]

M. Kunze and J. van Neerven, Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations, Journal of Differential Equations, 253 (2012), 1036-1068.  doi: 10.1016/j.jde.2012.04.013.  Google Scholar

[11]

A. Lipton, U. Pesavento and M. Sotiropoulos, Trading strategies via book imbalance, Risk, 70. Google Scholar

[12]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16, Springer Science & Business Media, 1995.  Google Scholar

[13]

A. Lunardi, Interpolation Theory, , Edizioni della Normale, Pisa, 2009.  Google Scholar

[14]

M. Müller, A stochastic Stefan-type problem under first-order boundary conditions, The Annals of Applied Probability, 28 (2018), 2335-2369.  doi: 10.1214/17-AAP1359.  Google Scholar

[15]

J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Wien. Ber., 98 (1888), 965-983.   Google Scholar

[16]

J. Van NeervenM. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces, Journal of Functional Analysis, 255 (2008), 940-993.  doi: 10.1016/j.jfa.2008.03.015.  Google Scholar

[17]

Z. Zheng, Stochastic Stefan Problems: Existence, Uniqueness, and Modeling of Market Limit Orders, Ph.D thesis, University of Illinois at Urbana-Champaign, 2012.  Google Scholar

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