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Greedy quantization: new approach and applications to reflected backward SDE

Abstract : This thesis contains two parts in which we treat greedy vector quantization with some financial applications. In the first part, we focus on greedy vector quantization. We start by presenting new theoretical and numerical approaches of greedy quantization. We establish new rate optimality results for a larger class of distributions, and carry out an extensive numerical study bringing many improvements in the greedy quantization-based numerical integration field. Among these studies, we present interesting numerical properties of greedy quantization sequences allowing them to become an advantageous component compared to sequences used in other numerical integration methods, like the low discrepancy sequences in the quasi-Monte Carlo method for example. Furthermore, we show that, when an Lr-optimal greedy quantization sequence is dilated or contracted in an appropriate way, it remains Ls-rate optimal. This is sometimes conditioned by a certain moment assumption on the underlying probability distribution. The second part of this manuscript is devoted to the approximation of a reflected Backward Stochastic Differential Equation by vector quantization. First, we establish upper bounds for the Lp-error, p 2 (1, 2+d), induced by recursive quantization of a general Markov chain one the one hand, and by a kind of “hybrid” recursive quantization, a method introduced in this thesis, on the other hand. Then, we establish Lp-error bounds, p 2 (1, 2+d), for the quantization-based space discretization scheme corresponding to the reflected Backward Stochastic Differential Equation. This is used for pricing financial options, mainly American options, and illustrated in several examples where we compare the behavior of recursive quantization versus greedy quantization in terms of precision and time cost. We use this discretization technique for the pricing of Barrier options as well.
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https://tel.archives-ouvertes.fr/tel-03103986
Contributor : Rancy EL NMEIR Connect in order to contact the contributor
Submitted on : Friday, January 8, 2021 - 3:19:41 PM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM
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  • HAL Id : tel-03103986, version 1

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Rancy El Nmeir. Greedy quantization: new approach and applications to reflected backward SDE. Probability [math.PR]. Sorbonne Université; Université Saint-Joseph de Beyrouth, 2020. English. ⟨tel-03103986⟩

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