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A note on the good lambda inequalities

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A note on the good lambda inequalities SÉMINAIRE DE PROBABILITÉS (STRASBOURG) SAUL D. JACKA A note on the good lambda inequalities Séminaire de probabilités (Strasbourg), tome 23 (1989), p. 57-65. <http://www.numdam.org/item?id=SPS_1989__23__57_0> © Springer-Verlag, Berlin Heidelberg New York, 1989, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A note on the good lambda inequalities S.D. Jacka Department of Statistics University of Warwick Coventry CV47AL UK §1. Introduction 1.1 The celebrated good-lambda inequalities of the form P(XT > YT ~a) > (,Q -1)a) and which are due to Burkholder (see Burkholder ( 1973), play a crucial role in deducing inequalities of the form CpllYTllp d optional T, or, more generally, EF(XT) CFEF(YT) V optional T (I.I.I) where X and Y are positive increasing previsible processes and F is a moderate function (see, for example,Azéma,Gundy and Yor(1980) Bass (1987), Davis (1987), Barlow and Yor (1982), and the seminal paper by Lenglart, Lepingle and Pratelli (1980)). Inequalities such as (1.1.1) are deduced by proving that the constant c(x,y;z), appearing in P(XT > x; YT y) _ c(x, y; z)P(XT > z) V optional T (1.1.2), has a suitable form. The main result of this paper is Theorem 5. If X is right continuous and previsible then if x > z > Xo, the best constant appearing in (1.1.2) is ya z) _ x~ YS:r: where Su = inf{t > 0 : Xt > u}. 1.2 Many interesting processes in martingale theory satisfy the conditions of Theorem 5 and are time-ch

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