{'fraction_non_alphanumeric': 0.013240857503152586, 'fraction_numerical': 0.0006305170239596469, 'mean_word_length': 5.151162790697675, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Martingales with specified marginals

This thesis provides methods for constructing martingales with specified marginals. The first collection of methods proceeds by quantization. This means that a measure is approximated by another measure whose support consits in a finite number of points. We introduce a quantization method which preserves the convex order. The convex order is a partial order on the space of measures which compares measures in terms of their relative dispersion. This new quantization method has the advantage that if two measures admit a martingale transition, then the quantized measures admit a martingale transition as well. This is not the case for the commonly used quantization method, the L2 quantization. For the quantized measures we present several methods to construct martingale transitions. The first method proceeds by linear programming. The second method proceeds by constructing matrices with specified diagonal and spectrum. The third method uses the Chacon Walsh algorithm. In a second part this thesis presents a new solution to the Skorokhod embedding problem. In a third part, this thesis studies the construction of continuous time martingales with specified marginals. Constructions are given using the Brownian Sheet. Other constructions are given by modifying a method developed by Albin. Martingales constructed in such a way have a scaling property. In a final part we establish some consequences of this theory with regards to the management of the risk of Asian options, with respect to their sensitivity to volatility and maturity.