On the denseness of the subset of discrete distributions in a certain set of two-dimensional distributions
Volume 9, Issue 3 (2022), pp. 265–277
Pub. online: 25 March 2022
Type: Research Article
Open Access
Open Access
Received
1 December 2021
1 December 2021
Accepted
16 March 2022
16 March 2022
Published
25 March 2022
25 March 2022
Abstract
In the article [Theory of Probability & Its Applications 62(2) (2018), 216–235], a class $\mathbb{W}$ of terminal joint distributions of integrable increasing processes and their compensators was introduced. In this paper, it is shown that the discrete distributions lying in $\mathbb{W}$ form a dense subset in the set $\mathbb{W}$ for ψ-weak topology with a gauge function ψ of linear growth.
References
Cox, A.M.G., Kinsley, S.M.: Discretisation and duality of optimal Skorokhod embedding problems. Stoch. Process. Appl. 129(7), 2376–2405 (2019). MR3958436. https://doi.org/10.1016/j.spa.2018.07.008
Föllmer, H., Schied, A.: Stochastic Finance. An introduction in discrete time. De Gruyter Graduate. Fourth revised and extended edition of [MR1925197]. De Gruyter, Berlin (2016). 596 pp. MR3859905. https://doi.org/10.1515/9783110463453
Gushchin, A.A.: The joint law of terminal values of a nonnegative submartingale and its compensator. Theory Probab. Appl. 62(2), 216–235 (2018). MR3649035. https://doi.org/10.1137/S0040585X97T988575
Gushchin, A.A.: Single jump filtrations and local martingales. Mod. Stoch. Theory Appl. 7(2), 135–156 (2020). MR4120612
Gushchin, A.A.: The joint law of a max-continuous local submartingale and its maximum. Theory Probab. Appl. 65(4), 545–557 (2021). MR4167880. https://doi.org/10.4213/tvp5339
Resnick, S.I.: Heavy-tail Phenomena. Probabilistic and statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer (2007), 404 pp. MR2271424
Rogers, L.C.G.: The joint law of the maximum and terminal value of a martingale. Probab. Theory Relat. Fields 95(4), 451–466 (1993). MR1217446. https://doi.org/10.1007/BF01196729
