Center for Computational Finance and Economic Systems (CoFES)

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Browsing Center for Computational Finance and Economic Systems (CoFES) by Subject "Applied mathematics"

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    An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Differential Equations with Uncertain Coefficients
    (2012-09-05) Kouri, Drew; Heinkenschloss, Matthias; Sorensen, Danny C.; Riviere, Beatrice M.; Cox, Dennis D.
    Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.
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    Modeling price dynamics on electronic stock exchanges with applications in developing automated trading strategies
    (2009) Gershman, Darrin Matthew; Riedi, Rudolf H.
    This thesis develops models for accurate prediction of price changes on electronic stock exchanges by utilizing autoregressive and logistic methods. Prices on these electronic stock exchanges, also called ECNs, are solely determined by where orders have been placed into the order book, unlike traditional stock exchanges where prices are determined by an expert market maker. Identifying the significant variables and formulating the models will provide critical insight into the dynamics of prices on ECNs. Whereas previous research has relied on simulated data to test market strategies, this analysis will utilize actual ECN data. The models recognize patterns of asymmetry and movement of the shares in the order book to formulate accurate probabilities for possible future price changes. On traditional stock exchanges, price changes could only occur as quickly as human beings could enact them. On ECNs, computerized systems place orders on behalf of traders based on their preferences, resulting in price changes that reflect trader activity almost instantaneously. The quickness of this automation on ECNs forces the re-evaluation of commonly held beliefs about stock price dynamics. Previous strategies developed for trading on ECNs have relied mainly on price fluctuations to gain profits. This thesis uses the formulated models to design profitable strategies that use accurate prediction rather than price variability.
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    Optimization governed by stochastic partial differential equations
    (2010) Kouri, Drew P.; Heinkenschloss, Matthias
    This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations.