Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times

Abstract With the help of the Garsia-Rodemich-Rumsey lemma, some improvements of the Burkholder-Davis-Gundy inequalities for continuous martingales are obtained, as well as certain estimates of the L p norm of the supremum of (a) the local times of a continuous semi-martingale and (b) the difference of the local times of two continuous martingales.

[1]  P. Meyer.  Flot d'une equation differentielle stochastique (d'après malliavin, bismut, kunita) , 1981 .

[2]  B. Davis,et al.  On the $L^p$ norms of stochastic integrals and other martingales , 1976 .

[3]  A. Garsia.  The Burgess Davis inequalities via Fefferman's inequality , 1973 .

[4]  D. Burkholder.  Distribution Function Inequalities for Martingales , 1973 .

[5]  A Sharp Inequality for Martingale Transforms , 1979 .

[6]  J. Neveu.  Bases mathématiques du calcul des probabilités , 1966 .

[7]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[8]  L. Dubins,et al.  ON CONTINUOUS MARTINGALES. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[9]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[10]  A. A. Novikov.  On moment inequalities and identities for stochastic integrals , 1973 .

[11]  Application de la relation de domination a certains renforcements des inegalites de martingales , 1982 .

[12]  M. Yor,et al.  (Semi-) martingale inequalities and local times , 1981 .

[13]  D. Ray.  Sojourn times of diffusion processes , 1963 .

[14]  J. M. Clark,et al.  The Representation of Functionals of Brownian Motion by Stochastic Integrals , 1970 .

[15]  D. Lépingle,et al.  Présentation unifiée de certaines inégalités de la théorie des martingales , 1980 .

[16]  D. Burkholder,et al.  Extrapolation and interpolation of quasi-linear operators on martingales , 1970 .

[17]  A. Garsia.  Combinatorial inequalities and smoothness of functions , 1976 .

[18]  F. Knight,et al.  Random walks and a sojourn density process of Brownian motion , 1963 .

[19]  Adriano M. Garsia,et al.  A Real Variable Lemma and the Continuity of Paths of Some Gaussian Processes , 1970 .

[20]  A local time inequality for martingales , 1983 .

[21]  D. Lépingle,et al.  La variation d'ordre p des semi-martingales , 1976 .

[22]  E. Lenglart,et al.  Relation de domination entre deux processus , 1977 .

[23]  Alexander Novikov,et al.  On Stopping Times for a Wiener Process , 1971 .