A generalisation of the Burkholder-Davis-Gundy inequalities

arXiv (Cornell University), 2020

For a continuous L2-bounded Martingale with no intervals of constancy, starting at 0 and having final variance σ 2 , the expected local time at x ∈ R is at most √ σ 2 + x 2 − |x|. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval (x− √ σ 2 + x 2 , x+ √ σ 2 + x 2). Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals, have been established by Dubins and Schwarz (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).

Journal of Multivariate Analysis, 1986

2000

Let (t )a nd(t) be nonnegative convex functions, and let ' and be the right continuous derivatives of and ; respectively. In this paper, we prove the equivalence of the following three conditions: (i)kfk ckfk; (ii) L H and (iii) Rt s0 '(s) s ds c (ct); 8 t>s 0; where L and H are the Orlicz martingale spaces. As

matematica.ciens.ucv.ve

Given a probability space (Ω,F,P) and a filtration {Fn}n≥1 of F, let us consider X = {Xn}n≥1 be {F n }- martingale and Y = {Yn}n≥1 a sequence of L1-bounded random variables, {Fn}-adapted, that we will call a multiplier sequence. Suppose that X = {Xn}n≥1 is a ...

Probability Theory and Related Fields, 1990

Let (f,) be a martingale. We establish a relationship between exponential bounds for the probabilities of the type P(I f.I > 2']1 T(f,)II ~) and the size of the constant Cp appearing in the inequality ]] f* Imp < c, iF T* (f)lip, for some quasi-linear operators acting on martingales.

2012

We present a unified approach to Doob&#39;s $L^p$ maximal inequalities for $1\leq p&lt;\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary \emph{deterministic} counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob&#39;s maximal inequalities. These are best possible in the

The Annals of Probability

The following conditions are necessary and sufficient for an arbitrary càdlàg local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in L 1 , on the events that those times are finite; and (C) its almost sure limit is an integrable random variable.

Lecture Notes in Mathematics, 1997

On the tails of the supremum and the quadratic variation of strictly local martingales

Electronic Communications in Probability, 2022

The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair (x, y) of nonnegative numbers satisfying x p ≤ y, we identify the optimal upper bounds for | sup n fn| p,∞, for nonnegative martingales f = (fn) n≥0 satisfying f 1 = x and f p p = y.

Proceedings of the American Mathematical Society, 1994

Let ( f n ) ({f_n}) and ( g n ) ({g_n}) be two martingales with respect to the same filtration ( F n ) ({\mathcal {F}_n}) such that their difference sequences ( d n ) ({d_n}) and ( e n ) ({e_n}) satisfy \[ P ( d n ≥ λ | F n − 1 ) = P ( e n ≥ λ | F n − 1 ) P({d_n} \geq \lambda |{\mathcal {F}_{n - 1}}) = P({e_n} \geq \lambda |{\mathcal {F}_{n - 1}}) \] for all real λ \lambda ’s and n ≥ 1 n \geq 1 . It is known that \[ ‖ f ∗ ‖ p ≤ K p ‖ g ∗ ‖ p , 1 ≤ p &gt; ∞ , {\left \| {{f^ \ast }} \right \|_p} \leq {K_p}{\left \| {{g^ \ast }} \right \|_p},\quad 1 \leq p &gt; \infty , \] for some constant K p {K_p} depending only on p. We show that K p = O ( p ) {K_p} = O(p) . This will be obtained via a new version of Rosenthal’s inequality which generalizes a result of Pinelis and which may be of independent interest.