Analyse d'Algorithmes Stochastiques Appliqués à la Finance

Abstract : This thesis is about stochastic approximation analysis and application in Finance. In the first part, a convergence result for stochastic approximation where the innovations satisfy averaging assumptions with some rate is established. It is applied to different types of innovations and illustrated on examples mainly motivated by Finance. A result on "universal" rate of convergence is then presented when the innovations are uniformly distributed and compared to those obtained in the i.i.d. framework. The second part is devoted to applications. First an optimal allocation problem applied to dark pools is studied. The execution of the maximum of the desired quantity leads to the design of a constrained stochastic algorithm studied in the i.i.d. and averaging frameworks. The next chapter presents a constrained optimization stochastic algorithm with projection to find the optimal posting distance in a limit order book by minimizing the execution cost of a given quantity. Parameter implicitation and calibration in financial models using stochastic approximation are then studied and illustrated by examples of applications on Black-Scholes, Merton and pseudo-CEV models. The last chapter is about stochastic approximation application to randomized urn models used in clinical trials. Thanks to ODE and SDE methods, the consistency and asymptotic normality results of Bai and Hu are retrieved under less stringent assumptions on the generating matrices.
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Submitted on : Thursday, December 15, 2011 - 12:21:55 AM
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Sophie Laruelle. Analyse d'Algorithmes Stochastiques Appliqués à la Finance. Probabilités [math.PR]. Université Pierre et Marie Curie - Paris VI, 2011. Français. ⟨tel-00652128⟩

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