The Annals of Applied Probability

A stochastic Stefan-type problem under first-order boundary conditions

Marvin S. Müller

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Abstract

Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and nonlinear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which introduces additional challenges in the analysis. Working on the distribution space, the Itô–Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochastic evolution equations and stochastic maximal $L^{p}$-regularity, we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary interaction even when the coefficients of the original problem have linear growths.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2335-2369.

Dates
Received: August 2016
Revised: July 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780275

Digital Object Identifier
doi:10.1214/17-AAP1359

Mathematical Reviews number (MathSciNet)
MR3843831

Zentralblatt MATH identifier
06974753

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 91B70: Stochastic models 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Stochastic partial differential equations Stefan problem moving boundary problem limit order book

Citation

Müller, Marvin S. A stochastic Stefan-type problem under first-order boundary conditions. Ann. Appl. Probab. 28 (2018), no. 4, 2335--2369. doi:10.1214/17-AAP1359. https://projecteuclid.org/euclid.aoap/1533780275


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References

  • [1] Amann, H. (2009). Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces. Jindřich Nečas Center for Mathematical Modeling Lecture Notes 6. Matfyzpress, Prague.
  • [2] Antonopoulou, D. C., Blömker, D. and Karali, G. D. (2015). The sharp interface limit for the stochastic Cahn–Hilliard equation. Available at arXiv:1507.07469.
  • [3] Auscher, P., McIntosh, A. and Nahmod, A. (1997). The square root problem of Kato in one dimension, and first order elliptic systems. Indiana Univ. Math. J. 46 659–695.
  • [4] Barbu, V. and Da Prato, G. (2002). The two phase stochastic Stefan problem. Probab. Theory Related Fields 124 544–560.
  • [5] Bayer, C., Horst, U. and Qiu, J. (2017). A functional limit theorem for limit order books with state dependent price dynamics. Ann. Appl. Probab. 27 2753–2806.
  • [6] Biais, B., Hillion, P. and Spatt, C. (1995). An empirical analysis of the limit order book and the order flow in the Paris bourse. J. Finance 50 1655–1689.
  • [7] Bouchaud, J. P., Mézard, M. and Potters, M. (2002). Statistical properties of stock order books: Empirical results and models. Quant. Finance 2 251–256.
  • [8] Brzeźniak, Z. and Veraar, M. (2012). Is the stochastic parabolicity condition dependent on $p$ and $q$? Electron. J. Probab. 17 no. 56, 24.
  • [9] Cont, R. (2011). Statistical modeling of high-frequency financial data. IEEE Signal Processing Magazine 28 16–25.
  • [10] Cont, R., Kukanov, A. and Stoikov, S. (2014). The price impact of order book events. J. Financ. Econom. 12 47–88.
  • [11] Cont, R., Stoikov, S. and Talreja, R. (2010). A stochastic model for order book dynamics. Oper. Res. 58 549–563.
  • [12] Da Prato, G., Kwapień, S. and Zabczyk, J. (1987). Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23 1–23.
  • [13] Da Prato, G. and Zabczyk, J. (2014). Stochastic Equations in Infinite Dimensions, 2nd ed. Encyclopedia of Mathematics and Its Applications 152. Cambridge Univ. Press, Cambridge.
  • [14] Donier, J., Bonart, J., Mastromatteo, I. and Bouchaud, J.-P. (2015). A fully consistent, minimal model for non-linear market impact. Quant. Finance 15 1109–1121.
  • [15] Gould, M. D., Porter, M. A., Williams, S., McDonald, M., Fenn, D. J. and Howison, S. D. (2013). Limit order books. Quant. Finance 13 1709–1742.
  • [16] Grisvard, P. (1967). Caractérisation de quelques espaces d’interpolation. Arch. Ration. Mech. Anal. 25 40–63.
  • [17] Haase, M. (2006). The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications 169. Birkhäuser, Basel.
  • [18] Huang, W., Lehalle, C.-A. and Rosenbaum, M. (2015). Simulating and analyzing order book data: The queue-reactive model. J. Amer. Statist. Assoc. 110 107–122.
  • [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
  • [20] Keller-Ressel, M. and Müller, M. S. (2016). A Stefan-type stochastic moving boundary problem. Stoch. Partial Differ. Equ. Anal. Computat. 4 no. 4, 746–790.
  • [21] Kim, K., Mueller, C. and Sowers, R. B. (2010). A stochastic moving boundary value problem. Illinois J. Math. 54 927–962.
  • [22] Kim, K., Zheng, Z. and Sowers, R. B. (2012). A stochastic Stefan problem. J. Theoret. Probab. 25 1040–1080.
  • [23] Kirilenko, A. A., Kyle, A. S., Samadi, M. and Tuzun, T. (2011). The flash crash: The impact of high frequency trading on an electronic market. Social Science Research Network Working Paper Series.
  • [24] Krylov, N. V. (1999). An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr. 64 185–242. Amer. Math. Soc., Providence, RI.
  • [25] Krylov, N. V. (2011). On the Itô–Wentzell formula for distribution-valued processes and related topics. Probab. Theory Related Fields 150 295–319.
  • [26] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
  • [27] Lions, J. L. and Magenes, E. (1972). Non-homogeneous Boundary Value Problems and Applications—1. Springer, Berlin.
  • [28] Lipton, A., Pesavento, U. and Sotiropoulos, M. G. (2014). Trading strategies via book imbalance. Risk.
  • [29] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel.
  • [30] Lunardi, A. (2004). An introduction to parabolic moving boundary problems. In Functional Analytic Methods for Evolution Equations (M. Iannelli, R. Nagel and S. Piazzera, eds.). Lecture Notes in Mathematics 1855 371–399. Springer, Berlin Heidelberg.
  • [31] Lunardi, A. (2009). Interpolation Theory, 2nd ed. Edizioni della Normale, Pisa.
  • [32] Markowich, P. A., Teichmann, J. and Wolfram, M. T. (2016). Parabolic free boundary price formation models under market size fluctuations. Multiscale Model. Simul. 14 1211–1237.
  • [33] Mastromatteo, I., Tóth, B. and Bouchaud, J.-P. (2014). Anomalous impact in reaction-diffusion financial models. Phys. Rev. Lett. 113 268701.
  • [34] Mueller, M. S. (2016). Semilinear stochastic moving boundary problems. Doctoral thesis, TU Dresden.
  • [35] Smith, E., Farmer, J. D., Gillemot, L. and Krishnamurthy, S. (2003). Statistical theory of the continuous double auction. Quant. Finance 3 481–514.
  • [36] Stefan, J. (1888). Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Wien. Ber. XCVIII, Abt. 2a (965–983).
  • [37] Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18. North-Holland, Amsterdam.
  • [38] van Neerven, J., Veraar, M. and Weis, L. (2012). Maximal $L^{p}$-regularity for stochastic evolution equations. SIAM J. Math. Anal. 44 1372–1414.
  • [39] van Neerven, J., Veraar, M. and Weis, L. (2012). Stochastic maximal $L^{p}$-regularity. Ann. Probab. 40 788–812.
  • [40] Zheng, Z. (2012). Stochastic Stefan problems: Existence, uniqueness and modeling of market limit orders. Ph.D. thesis, Graduate College of the Univ. Illinois at Urbana-Champaign.