Webmartingale in plain sight, since one can construct it out of thin air. 2 Ville’s and Doob’s inequalities The rst of Doob’s inequalities can be seen as a uniform generalization of … WebDec 24, 2024 · We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue …
Doob’s maximal inequalities for martingales in variable
WebInequality ( 1) is also known as Kolmogorov’s submartingale inequality. Doob’s inequalities are often applied to continuous-time processes, where T =R+ 𝕋 = ℝ +. In this case, X∗ t = sups≤t Xs X t * = sup s ≤ t X s is a supremum of uncountably many random variables, and need not be measurable. Instead, it is typically assumed ... rmb money maximiser
Math 280B, Winter 2005 - University of California, …
WebOct 1, 2024 · 1.2. The main result. In this paper we prove the analogue result of Theorem 1.2 in the case when and as a consequence we get the variant of the classical Doob’s maximal inequality. Let , for all x > 0 and 1 < p < ∞. Then, we can easily see that δ p is strictly convex function on the interval 0, 2 p − 1 p − 1 and strictly concave ... In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time. As the name suggests, the result is … See more The setting of Doob's inequality is a submartingale relative to a filtration of the underlying probability space. The probability measure on the sample space of the martingale will be denoted by P. The corresponding See more Doob's inequality for discrete-time martingales implies Kolmogorov's inequality: if X1, X2, ... is a sequence of real-valued independent random variables, each with mean … See more • Shiryaev, Albert N. (2001) [1994], "Martingale", Encyclopedia of Mathematics, EMS Press See more There are further submartingale inequalities also due to Doob. Now let Xt be a martingale or a positive submartingale; if the index set is uncountable, then (as above) assume that the sample paths are right-continuous. In these scenarios, See more Let B denote canonical one-dimensional Brownian motion. Then The proof is just as … See more Webwhere the last one is Jensen’s inequality. Theorem 29 (Doob’s decomposition) If (Xn, Bn)n∗0 is a submartingale then it can be uniquely decomposed as Xn = Zn + Yn, where (Yn, Bn) is martingale, Z0 = 0,Zn → Zn+1 almost surely and Zn is Bn−1-measurable. Proof. Let Dn = Xn − Xn−1 and Gn = E(Dn Bn−1) = E(Xn Bn−1) − Xn−1 ∗ 0 rmblsp