On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales

Choulli, Tahir; Stricker, Christoph; Krawczyk, Leszek

doi:10.1007/s004400050218

Cited by 18 publications

(10 citation statements)

References 15 publications

(20 reference statements)

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“…By the weighted Burkholder-Davis-Gundy inequality [see Choulli, Krawczyk and Stricker (1997) for a recent treatment], the fact that p − 1 = p/q together with the estimate…”

Section: Proof It Was Proved Inmentioning

confidence: 99%

On the minimal entropy martingale measure

Grandits¹,

Rheinländer²

2002

Ann. Probab.

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Let X be a locally bounded semimartingale. Using the theory of BMOmartingales we give a sufficient criterion for a martingale measure for X to minimize relative entropy among all martingale measures. This is applied to prove convergence of the q-optimal martingale measure to the minimal entropy martingale measure in entropy for q ↓ 1 under the assumption that X is continuous and that the density process of some equivalent martingale measure satisfies a reverse LLogL-inequality.

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“…By the weighted Burkholder-Davis-Gundy inequality [see Choulli, Krawczyk and Stricker (1997) for a recent treatment], the fact that p − 1 = p/q together with the estimate…”

Section: Proof It Was Proved Inmentioning

confidence: 99%

On the minimal entropy martingale measure

Grandits¹,

Rheinländer²

2002

Ann. Probab.

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“…The following lemma, that plays crucial role in our estimations, ia interesting in itself and generalizes [15,Lemma 4.8] . Proof.…”

Section: Appendix a Some Martingale Inequalitiesmentioning

confidence: 78%

“…At the methodical aspect, we elaborate our prior estimates using different method than the existing ones in the literature. Indeed, we directly establish inequalities without distinguishing the cases on p, and this is due to some stronger and deeper martingales inequalities of [15] that we slightly generalize. Furthermore, our method is robust towards the nature of the filtration F, and hence our analysis can be extended to setting with jumps without serious difficulties.…”

Section: Main Challenges and Our Achievementsmentioning

confidence: 99%

Reflected backward stochastic differential equations under stopping with an arbitrary random time

Alsheyab,

Choulli

2021

Preprint

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This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of   Here τ is an arbitrary random time that might not be a stopping time for the filtration F generated by the Brownian motion W . We consider the filtration G resulting from the progressive enlargement of F with τ where this becomes a stopping time, and study the RBSDE under G. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data (f, ξ, S, τ ) that guarantee the existence of the solution of the G-RBSDE in L p (p > 1)? b) How can we estimate the solution in norm using the triplet-data (f, ξ, S)? c) Is there an RBSDE under F that is intimately related to the current one and how their solutions are related to each other? This paper answers all these questions deeply and beyond. Importantly, we prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor E -a positive and non-increasing F-adapted and RCLL process-that is vital in answering our questions without assuming any further assumption on τ , and determining the space for the triplet-data (f, ξ, S) and the space for the solution of the RBSDE as well. Furthermore, we found that the conditions for the G-RBSDE are weaker that the conditions for its F-RBSDE counterpart when the horizon is unbounded. Our approach sounds novel and very robust, as it relies on sharp martingale inequalities that hold no matter what is the filtration, and it treats both the linear and general case of RBSDEs for bounded and unbounded horizon. * Tahir Choulli is grateful to Polytechnique for the hospitality, where this work started in March 2018, and he is grateful to Nizar Touzi for introducing him to the RBSDEs and proposing him this project and its main ideas.

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“…We refer to [DMSSS97] for a detailed presentation of these topics for the above discussed closedness of G in L p for p = 2 and the case of R d -valued continuous semi-martingales S. Extensions to the case 1 < p < ∞ as well as to the case of general R d -valued semi-martingales were obtained in [GK98], [CKS97] and [CKS98].…”

Section: Weighted Norm Inequalities and Closedness Of A Space Of Stocmentioning

confidence: 99%

Applications to Mathematical Finance

Delbaen

Schachermayer

2001

Handbook of the Geometry of Banach Spaces

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We give an introduction to the theory of Mathematical Finance with special emphasis on the applications of Banach space theory.The introductary section presents on an informal and intuitive level some of the basic ideas of Mathematical Finance, in particular the notions of "No Arbitrage" and "equivalent martingale measures". In section two we formalize these ideas in a mathematically rigorous way and then develop in the subsequent four sections some of the basic themes.Of course, in this short handbook-contribution we are not able to give a comprehensive overview of the whole field of Mathematical Finance; we only concentrate on those issues where Banach space theory plays an important role.

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On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales

Cited by 18 publications

References 15 publications

On the minimal entropy martingale measure

On the minimal entropy martingale measure

Reflected backward stochastic differential equations under stopping with an arbitrary random time

Applications to Mathematical Finance

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