http://galton.uchicago.edu/~wbwu/papers/lp31aug.pdf WebOct 1, 2024 · Doob’s maximal inequalities Martingales 1. Introduction 1.1. Doob’s inequalities Let be a probability space. If is a random variable, then the expectation of is denoted by with respect to the probability .
Math 280B, Winter 2005 - University of California, …
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 usually given … 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 $${\displaystyle P\left[\sup _{0\leq t\leq T}B_{t}\geq C\right]\leq \exp \left(-{\frac {C^{2}}{2T}}\right).}$$ See more Webwhere the inequality follows from the submartingale property of Y because A∩{T= n}∈F n. 3. Proofofthe maximal inequality.Fix a positive integer nand define T:= min{k≥ 0: X k ≥ … k9 cipher\\u0027s
Doob
WebTHEOREM AND THE DOOB’S MARTINGALE INEQUALITY JINGGUO LAI Abstract. Evaluation of the Bellman functions is a difficult task. The exact Bellman func-tions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work for the tree-like structure. WebTHE MAXIMUM MAXIMUM OF A MARTINGALE 3 equality in our inequality. We note that the idea to derive martingale in-equalities from pathwise inequalities was pivotal to the pioneering work on robust pricing and hedging of Hobson [25] and was recently underlined in Acciaio et al. [1]. Mathematical finance motivation. The problem we consider, as ... Weba generalization of the well-known Doob maximal inequality for martingales to adapted sequences. It states that the moment of order p of the maximum of the partial sums of … k9 cliff\\u0027s