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Optimal quantization methods with applications to finance.

Abstract : THIS THESIS IS DOVOTED TO OPTIMAL QUANTIZATION WITH SOME APPLICATIONS TO MATHEMATICAL FINANCE. CHAP.1 REMINDS THE BASES OF OPTIMAL QUANTIZATION AND NUMERICAL SEARCH OF OPTIMAL QUANTIZERS. IN CHAP.2 WE STUDY THE ASYMPTOTICS, IN L^S, OF THE QUANTIZATION ERROR ASSOCIATED TO A LINEAR TRANSFORM OF AN L^R OPTIMAL SEQUENCE OF QUANTIZERS. WE SHOW THAT SUCH A TRANSFORMATION ALLOWS TO MAKE THE TRANSFORMED SEQUENCE L^S RATE OPTIMAL FOR EVERY S>0, FOR A LARGE FAMILY OF PROBABILITIES. CHAP.3 DEALS WITH THE ASYMPTOTICS OF THE MAXIMAL RADIUS SEQUENCE ASSOCIATED TO AN L^R OPTIMAL SEQUENCE OF QUANTIZERS. WE SHOW THAT AS SOON AS SUPP(P) IS UNBOUNDED, THE MAXIMAL RADIUS CONVERGE TO INFINITY. WE THEN GIVE THE RATE OF CONVERGENCE FOR A LARGE FAMILY OF PROBABILITIES. CHAP.4 IS DEVOTED TO THE PRICING OF LOOKBACK AND BARRIER LIKE OPTIONS. WE WRITE THESE PRICES IN A FORM WHICH ALLOWS US TO ESTIMATE THEM BY MONTE CARLO, BY AN HYBRID MONTE CARLO-QUANTIZATION AND BY A PUR QUANTIZATION METHOD.
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Contributor : Abass Sagna <>
Submitted on : Thursday, November 27, 2008 - 1:32:52 PM
Last modification on : Wednesday, December 9, 2020 - 3:13:14 PM
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  • HAL Id : tel-00342033, version 1

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Abass Sagna. Optimal quantization methods with applications to finance.. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2008. English. ⟨tel-00342033⟩

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