We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover, our deterministic inequalities lead to new versions of Doob's maximal inequalities. These are best possible in the sense that equality is attained by properly chosen martingales.
We present the formalization of Doob's martingale convergence theorems in the mathlib library for the Lean theorem prover. These theorems give conditions under which (sub)martingales converge, almost everywhere or in $L^1$. In order to formalize those results, we build a definition of the conditional expectation in Banach spaces and develop the theory of stochastic processes, stopping times and martingales. As an application of the convergence theorems, we also present the formalization of Lévy's generalized Borel-Cantelli lemma. This work on martingale theory is one of the first developments …
We characterize the random times $ρ$ whose Azema supermartingales $Z^ρ$ take the form $Z^ρ=U/U^*$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^*$ denotes the running maximum process of $U$.
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to[0,\infty)$ be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space $L^φ(Ω)$. Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on $L^φ(Ω)$. As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for $L^φ(Ω)$, which are new even in weighted Orlicz spaces. The authors then establish the atomic characterization…
We generalize a famous tail Doob's inequality, relative two non-negative random variables, arising in the martingale theory, in two directions: on the more general source data and on the random variables belonging to the so-called Grand Lebesgue Spaces. We bring also several examples in each sections in order to show the exactness of our estimates.
The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let $(\mathcal M,τ)$ be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras $(\mathcal M_t)_{0\leq t\leq1}$ whose union is weak-$*$ dense in $\mathcal{M}$. Let $\mathcal E_t$ denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for $1 < p < 2$ and $x \in L_p(\mathcal M,τ)$ one can find $a, b \in L_p(\mathcal M,τ)$ and contractions $u_t, v_t \in…
For a general adapted integrable right-continuous with left limits (RCLL) process $(X_t)_{t\in[0,τ]}$ taking values in a metric space $(\mathcal E,d)$, we show (among other things) that for every $m\in(1,\infty)$ $$ \frac{m-1}{2m-1}\|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m\le \|\sup_{t\in[0,τ]}d(X_0,X_t)\|_m\le c\frac{m^2}{m-1} \|\sup_{t\in[0,τ]}\mathbb{E}(d(X_{t-},X_τ)|\mathcal F_t)\|_m $$ with a universal constant $c$. This is a probabilistic version of Fefferman--Stein estimate for the sharp maximal functions. While the former inequality is derived easily from Doob's mart…
