Stochastic Stefan problems: existence, uniqueness, and modeling of market limit orders
Zheng, Zhi
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https://hdl.handle.net/2142/42335
Description
Title
Stochastic Stefan problems: existence, uniqueness, and modeling of market limit orders
Author(s)
Zheng, Zhi
Issue Date
2013-02-03T19:35:41Z
Director of Research (if dissertation) or Advisor (if thesis)
Sowers, Richard B.
Doctoral Committee Chair(s)
DeVille, Robert E.
Committee Member(s)
Sowers, Richard B.
Zharnitsky, Vadim
Rapti, Zoi
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
stochastic partial differential equations (PDEs)
moving boundaries
market limit orders
parameter estimation
Maximum-Likelihood Estimator (MLE)
Mean-Square Errors (MSE)
Akaike information criterion (AIC)
investment optimization
dynamic optimization
Abstract
In this thesis we study the effect of stochastic perturbations on moving boundary value PDE's with Stefan boundary conditions, or Stefan problems, and show the existence and uniqueness of the solutions to a number of stochastic equations of this kind. We also derive the space and time regularities of the solutions and the associated boundaries via Kolmogorov's Continuity Theorem in a defined normed space.
Moreover, we model the evolution of market limit orders in completely continuous settings using such equations, derive parameter estimation schemes using maximum likelihood and least mean-square-errors methods under certain criteria, and settle the investment optimization problem in both static and dynamic sense when taking the model as exogenous.
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