A trajectorial interpretation of Doob’s martingale inequalities

Acciaio, Beatrice; Beiglböck, Mathias; Penkner, Friedrich; Schachermayer, Walter; Temme, Johannes

doi:10.1214/12-aap878

Cited by 42 publications

(86 citation statements)

References 19 publications

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“…We argue that for sufficiently large n the terms EQ|f i (S (n) )−f i (S (1) )|, i = 1, ..., N and EQ|G(S (n) ) − G(S (1) )| are small. Indeed, as before the factQ ∈ Mκ ,L implies that EQ[ S (1) ) ] ≤Ĉ (where, recall the constantĈ from Definition 4.1) and so lim n→∞ EQ[ S (1) χ {K≥n} ] = 0. From Assumption 2.1 we get lim sup…”

Section: Asymptotical Analysis Of the Boundsmentioning

confidence: 85%

“…From Theorem 1 in Skorokhod (1976) and the fact that the random variables W ti+1 − W ti , i = 0, .., r − 1 are independent, it follows that we can find a sequence of measurable function g (1) i , g (2) i : R 2i−1 → R, i = 1, ..., r with the following property. The stochastic processes (adapted to the Brownian filtration) {Ṡ W ti } r i=0 and {M W ti } r i=0 which are given by the recursion relationṡ…”

Section: Thus the Inequality Eq[(q(smentioning

confidence: 99%

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Convex Duality with Transaction Costs

Dolinsky

Soner

2017

Mathematics of OR

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Abstract. Convex duality for two two different super-replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one of the problems considered is the model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super-replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost.

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Section: Asymptotical Analysis Of the Boundsmentioning

confidence: 85%

Section: Thus the Inequality Eq[(q(smentioning

confidence: 99%

Convex Duality with Transaction Costs

Dolinsky

Soner

2017

Mathematics of OR

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“…It is easy to see that this is not true in general if X is a (strictly positive) submartingale: it is enough to put n = 1, X 0 = ε, where ε > 0 is small enough, and X 1 = 1. In other words, inequality (Doob-L 1 ) in the statement of Theorem 1.1 in [1] is valid for nonnegative martingales and is not valid for submartingales as is stated in this theorem. Nevertheless, the following improvement of Doob's maximal L log L-inequality is true: for any nonnegative submartingale X,…”

Section: Theoremmentioning

confidence: 91%

“…Inequality (5) was obtained in [1]. It implies the following minor generalization of Doob's maximal L p -inequality: if X is a nonnegative submartingale and EX…”

Section: Theoremmentioning

confidence: 99%

On pathwise counterparts of Doob’s maximal inequalities

Gushchin

2014

Proc. Steklov Inst. Math.

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In this paper, we present pathwise counterparts of Doob's maximal inequalities (on the probability of exceeding a level) for submartingales and supermartingales.Recently a new method of proving martingale inequalities became popular. Namely, they are derived (and thus often improved) from elementary deterministic inequalities. The purpose of this note is to present elementary pathwise counterparts of Doob's maximal inequalities on the probability of exceeding a level. Substituting a trajectory of a stochastic process in our inequality and taking expectations, we obtain Doob's inequality for supermartingales and submartingales, see [7, Chapter VII, Theorem 3.2], due to the fact that one of the terms in the inequalities can be dropped if the process is a supermartingale (in the case of the first inequality) or a submartingale (in the case of the second one). We also show that the pathwise counterparts of Doob's maximal L p -and L log L-inequalities from the paper [1] can be obtained by integration from our inequalities.

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“…This leads us to a similar structure to that in [19] and in other papers [6,8,10,11,14,15,18,21,22,23,24,25,29] which consider modelindependent pricing. This approach is very closely related to path-wise proofs of well-known probabilistic inequalities [2,9]. Apart from the continuity of the price process no other model assumptions are placed on the dynamics of the price process.…”

Section: Introductionmentioning

confidence: 99%

Martingale Optimal Transport and Robust Hedging in Continuous Time

Dolinsky

Soner

2013

SSRN Journal

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Abstract. The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.

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A trajectorial interpretation of Doob’s martingale inequalities

Cited by 42 publications

References 19 publications

Convex Duality with Transaction Costs

Convex Duality with Transaction Costs

On pathwise counterparts of Doob’s maximal inequalities

Martingale Optimal Transport and Robust Hedging in Continuous Time

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