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Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread

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  • Antonopoulou, D.C.
  • Farazakis, D.
  • Karali, G.

Abstract

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase [s−(t),s+(t)] defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread s+(t)−s−(t) stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green’s function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density.

Suggested Citation

  • Antonopoulou, D.C. & Farazakis, D. & Karali, G., 2025. "Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread," Stochastic Processes and their Applications, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:spapps:v:179:y:2025:i:c:s030441492400214x
    DOI: 10.1016/j.spa.2024.104506
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    References listed on IDEAS

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    1. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    2. repec:eme:sef000:sef-04-2014-0075 is not listed on IDEAS
    3. Michael Bleaney & Zhiyong Li, 2015. "The performance of bid-ask spread estimators under less than ideal conditions," Studies in Economics and Finance, Emerald Group Publishing Limited, vol. 32(1), pages 98-127, March.
    4. repec:bla:jfinan:v:44:y:1989:i:2:p:479-86 is not listed on IDEAS
    5. Michael Bleaney & Zhiyong Li, 2015. "The performance of bid-ask spread estimators under less than ideal conditions," Studies in Economics and Finance, Emerald Group Publishing Limited, vol. 32(1), pages 98-127, March.
    6. Cohen, Kalman J & Maier, Steven F & Schwartz, Robert A & Whitcomb, David K, 1981. "Transaction Costs, Order Placement Strategy, and Existence of the Bid-Ask Spread," Journal of Political Economy, University of Chicago Press, vol. 89(2), pages 287-305, April.
    7. Hasbrouck, Joel, 2004. "Liquidity in the Futures Pits: Inferring Market Dynamics from Incomplete Data," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 39(2), pages 305-326, June.
    8. Hambly, Ben & Kalsi, Jasdeep, 2020. "Stefan problems for reflected SPDEs driven by space–time white noise," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 924-961.
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