Maximal $\phi$-inequalities for nonnegative submartingales

Let (M n) n≥0 be a nonnegative submartingale and M * n def = max 0≤k≤n M k , n ≥ 0 the associated maximal sequence. For nondecreasing convex functions φ : [0, ∞) → [0, ∞) with φ(0) = 0 (Orlicz functions) we provide various inequalities for Eφ(M * n) in terms of EΦ a (M n) where, for a ≥ 0, Φ a (x) def = x a s a φ (r) r dr ds, x > 0. Of particular interest is the case φ(x) = x for which a variational argument leads us to EM * n ≤

Statistics & Probability Letters, 2008

New maximal inequalities for non-negative martingales are proved. The inequalities are tight and strengthen well-known maximal inequalities by Doob. The inequalities relate martingales to information divergence and imply convergence of X ln X bounded martingales. Similar results hold for stationary sequences.

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

We are continuing out studies of the so-called Markov inequalities with a majorant. Inequalities of this type provide a bound for the $k$-th derivative of an algebraic polynomial when the latter is bounded by a certain curved majorant $\mu$. A conjecture is that the upper bound is attained by the so-called snake-polynomial which oscillates most between $\pm \mu$, but it turned out to be a rather difficult question. In the previous paper, we proved that this is true in the case of symmetric majorant provided the snake-polynomial has a positive Chebyshev expansion. In this paper, we show that that the conjecture is valid under the condition of positive expansion only, hence for non-symmetric majorants as well.

2010

It is well known that if a submartingale $X$ is bounded then the increasing predictable process $Y$ and the martingale $M$ from the Doob decomposition $% X=Y+M$ can be unbounded. In this paper for some classes of increasing convex functions $f$ we will find the upper bounds for $\lim_n\sup_XEf(Y_n)$, where the supremum is taken over all submartingales $(X_n),0\leq X_n\leq 1,n=0,1,...$.

Stochastic Processes and their Applications, 1987

The well-known submartingale maximal inquality of Birnbaum and Marshall (1961) is generalized to provide upper tail inequalities for suprema of processes which are products of a submartingale by a nonincreasing nonnegative predictable process. The new inequalities are proved by applying an inequality of Lenglart (1977). and are then used to provide best-possible universal growth-rates for a general submartingale in terms of the predictable compensator of its positive part. Applications of these growth rates include strong asymptotic upper bounds on solutions to certain stochastic diticrential equations, and strong asymptotic lower bounds on Brownian-motion occup;Uion-times. A A/S Strbjecr C/a.cs(ficafmn.r: 6OG44, 60F IS.

2020

Consider a cadlag local martingale $M$ with square brackets $[M]$. In this paper, we provide lower and upper bounds for expectations of the type $E [M]^{q/2}_{\tau}$, for any stopping time $\tau$ and $q\ge 2$. This result is a Burkholder-Davis-Gundy-type inequality as it relates the expectation of the running maximum $|M^*|^q$ to the expectation of the dual previsible projections of the relevant powers of the associated jumps of $M$. The case of convex moderate functions is also treated.

Stochastic Processes and their Applications, 1988

An embedding of an arbitrary centred law p in a Brownian motion (that is a stopping time T and a Brownian motion B such that Z(B,) = p and (BlnT; t 2 0) is ui) is found such that I3F has a law which dominates that of MT, where the pair (M, T) is any other ui embedding of p in a martingale.

2012

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 \emph{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

Bulletin of the London Mathematical Society, 2005

We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory.