Beatrice Acciaio. Friedrich Penkner. Walter Schachermayer.

## Doob's martingale inequality

The Annals of Applied Probability , Vol. Penkner, By B. Acciaio, M. Beiglbo W. Schachermayer and J. 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. These are best possible in the sense that equality is attained by properly chosen martingales.

In this paper we derive estimates for the running max- imum of a martingale or nonnegative submartingale in terms of its terminal value. Among other results, we establish the following martingale inequalities. Theorem 1. Received March ; revised May AMS subject classifications.

## Sharp inequalities for martingales and stochastic integrals

Primary 60G42, 60G44; secondary 91G Key words and phrases. Doob maximal inequalities, martingale inequalities, pathwise hedging. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, , Vol. Trajectorial inequalities. The novelty of this note is that the above mar- tingale inequalities are established as consequences of deterministic counter- parts. We postpone the general statements Proposition 2. Let s0 ,. Inequalities in continuous time—sharpness. Dubins and Gilat [9] showed that the constant 2 in Doob-L2 is optimal, that is, cannot be replaced by a strictly smaller constant.

It is also natural to ask whether equality can be attained in Doob-L2. Keeping in mind that equality in Doob-L1 is attained, one may also try to improve on Doob-L2 by incorporating the starting value of the martingale. Indeed, we obtain the following result.

## The Khintchine inequalities and martingale expanding sphere of their action

In Theorem 3. We emphasize that the idea that Doob-Lp can be improved by in- corporating the starting value S0 into the inequality is not new.

Cox [7], Burkholder [5] and Peskir [18] show that 1. Here the constants 4 resp. However, she still faces the risk of S having its highest value at some time n before T. Alternatively, the sharpness of 1. The idea of robust pricing and pathwise hedging of exotic options seem- ingly goes back to Hobson [13] see also [4, 6, 15]. We refer the reader to [14] for a thorough introduction to the topic. Organization of the paper. We prove Theorem 3. Proof of Theorem 1. The proof of Theorem 1. Proposition 2.

A continuous time counterpart of Path-Lp is given in Remark 3. In the proof of Proposition 2. Lemma 2. This follows by properly rearranging the summands. In total, every summand on the left-hand side of 2. Proof of Proposition 2. Hence, Lemma 2. II By Lemma 2. By Proposition 2. Inequality Doob-L1 follows from Proposition 2.

### Florence Merlevède, Magda Peligrad, and Sergey Utev

As opposed to the strategies given in Proposition 2. Qualitative Doob Lp -inequality—Proof of Theorem 1. In this sec- tion we prove Theorem 1. Theorem 3. Then p 1 kS0 kpp 3.

Moreover, equality in 3. Remark 3.

Note that inequality 3. Connection between Theorems 1. We now discuss under which conditions Theorem 1. Recall that Theorem 1. Indeed, rearrang- ing 3. Definition of the continuous-time integral. For a general account on the theory of pathwise stochastic integration we refer to Bichteler [3] and Karandi- kar [16]. The standard argument of mixing sequences then implies uniqueness. We stress that 3. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.

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