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Pathwise Inequalities for Martingale-type Processes with Some Applications

Student: Nesterov Roman

Supervisor: Alexander A. Gushchin

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Bachelor)

Final Grade: 7

Year of Graduation: 2018

In the second section of the paper we derive a trajectory inequality that is applicable to trajectories of a nonnegative submartingale and a predictable increasing process from its decomposition of Doob. Next, we consider the restrictions imposed by the above inequality for the distributions of the random variables $ A_\infty, X_\infty $ under the assumption $ X_\infty - A_\infty = const \ne 0 $. If $ X_\infty - A_\infty $ is positive, then there exists a nontrivial boundary from below with respect to the stochastic order. It can be seen from the results of [1] that this boundary is reached. In the case where the constant is negative, the inequality imposes no restrictions. The aim of the paper is to show that there is no trivial lower bound for the distribution $ X_\infty $ under this assumption, namely for any nonnegative nonintegrable random variable $ Y $, we construct a nonnegative martingale $ M $ with $ \mathbb{E} (M) = 1 $ such that $ \sum_{k = 1} ^ \infty M_k $ is less than $ Y $ in the sense of stochastic order. Next, an increasing process $ X $ is constructed such that $ X_\infty = \sum_{k = 1}^\infty M_k $ and $ A_\infty = X_\infty + 1 $.

Full text (added June 3, 2018)

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