Extension of a stochastic Gronwall lemma

• Autorzy: Makasu Cloud

Identyfikatory

DOI

10.4064/ba190524-7-5

• Języki publikacji: EN

Abstrakty

EN

A stochastic Gronwall lemma is proved in Scheutzow (2013) in the case when the exponent p lies in the interval 0 < p <1. In this paper, we extend the lemma to the entire interval 0 < p < ∞. We construct simple examples to illustrate the present result.

Słowa kluczowe

EN

Burkholder martingale inequality; Gronwall lemma

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2020
• Tom: Vol. 68, no. 1
• Strony: 97--104
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Makasu Cloud

• Cloud Makasu Dept of Maths and Applied Maths, University of the Western Cape, Private Bag X17 Bellville 7535, Cape Town, South Africa

Bibliografia

• [1] R. Bañuelos and A. Osekowski, Sharp maximal Lp-estimates for martingales, Illinois J. Math. 58 (2014), 149-165.
• [2] J. M. Bismut and M. Yor, An inequality for processes which satisfy Kolmogorov’s continuity criterion. Application to continuous martingales, J. Funct. Anal. 51 (1983), 166-173.
• [3] D. L. Burkholder, One-sided maximal functions and Hp, J. Funct. Anal. 18 (1975), 429-454.
• [4] J. Y. Calais and M. Génin, Sur les martingales locales continues indexées par ]0;1[, in: Séminaire de probabilités (Strasbourg) XVII, Lecture Notes in Math. 986, Springer, 1983, 162-178.
• [5] H. J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (Part II), Math. Nachr. 144 (1989), 241-281.
• [6] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer, 1991.
• [7] P. Protter, Stochastic Integration and Differential Equations, Springer, 1990.
• [8] M. Scheutzow, A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Related Topics 16 (2013), art. 1350019, 4 pp.
• [9] D. Willet and J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality, Monatsh. Math. 69 (1965), 362-367.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).