Convergence of BS<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle></mml:math>Es driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver

Madan, Dilip B.; Pistorius, Martijn; Stadje, Mitja

doi:10.1016/j.spa.2015.11.013

Cited by 11 publications

(6 citation statements)

References 27 publications

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“…Under the bound in the right-hand side of (2.24) we have numerical stability of the solutions to sequence of BS∆Es driven by (L (π) ) (see [38,Theorem 3.4]).…”

Section: Convergencementioning

confidence: 99%

“…While one may prove the functional limit theorem directly through duality arguments, we present in the interest of brevity a proof that draws on the convergence results obtained in [38] for weak approximation of BSDEs. In the literature various related convergence results are available, of which we next mention a number (refer to [38] for additional references). The construction of continuoustime dynamic risk-measures arising as limits of discrete-time ones was studied in [46] in a Brownian setting.…”

Section: Introductionmentioning

confidence: 99%

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On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

2017

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In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

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“…Under the bound in the right-hand side of (2.24) we have numerical stability of the solutions to sequence of BS∆Es driven by (L (π) ) (see [38,Theorem 3.4]).…”

Section: Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

2017

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“…Remark 3.11. The last corollary generalizes Madan et al[50, Corollary 2.6]. Indeed, they consider a discretetime process approximating a Lévy process, and work with the natural filtrations, while we can deal both with discrete-and continuous-time approximations of general martingales, with arbitrary filtrations.3.3.Comparison with the literature.…”

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confidence: 69%

“…Mémin [52] looked into the stability of the canonical decomposition for semimartingales, Kchia [38] (see also Kchia and Protter [39]) extended Barlow and Protter's result [4] for stability of special semimartingale decompositions to a framework allowing changing filtrations, while Possamaï and Tan [61] extended results of [9,10] to the case of so-called second order BSDEs. Let us also mention the recent paper by Madan, Pistorius, and Stadje [50], which considers stability results for BSDEs with jumps (that is to say that both the processes Z and U are present in the solution), when the driving càdlàg martingale is approximated by random walks. Several of these works make a strong use of the notions of extended convergence, introduced by Aldous [1], as well as that of convergence of filtrations, introduced by Hoover [32] and further developed by Coquet, Mémin, and Mackevičius [16] and Coquet, Mémin, and Słomiński [17], which also plays a major role in the present paper.…”

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confidence: 99%

Stability results for martingale representations: The general case

Papapantoleon¹,

Possamaï²,

Saplaouras³

2019

Trans. Amer. Math. Soc.

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In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi-left-continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS A.3. Young functions 47 A.4. Proof of Corollary 3.4 51 REFERENCES 53 Once the semimartingales considered have more structural properties, other interesting results can be obtained. Barrieu, Cazanave, and El Karoui [6], for instance, and later Barrieu and El Karoui [5] were interested in what they coined "continuous quadratic semimartingales" (see also related articles by Mocha and Westray [54], still in the continuous case, and recent extensions to jump processes by Ngoupeyou [58] and El Karoui, Matoussi, and Ngoupeyou [25]), for which, roughly speaking, the bounded variation process part in the semimartingale X is absolutely continuous with respect to the quadratic variation of the martingale part of X.[5, 6] obtained associated stability results for these processes.An important common feature of the articles mentioned so far, is that they actually only consider the strong framework we described at the beginning of this introduction, in the sense that there is a always a fixed probability space and all processes (meaning here mainly X n and X ∞ ) are adapted to the same fixed filtration G. An important exception is Słomiński [67], where the probability space is fixed, but not the filtration. However, for practical purposes, and especially for the analysis of numerical schemes, it is well-known that the weak framework is also of paramount importance, as illustrated for instance by the famous Donsker theorem. There has thus been a certain number of studies of stability properties for semimartingale or martingale decompositions when the underlying filtration itself is also allowed to change. In that direction, Antonelli and Kohatsu-Higa [3], followed by Coquet, Mackevičius, and Mémin [14, 15], Ma, Protter, San Martín, and Torres [47], Briand, Delyon, and Mémin [9], Briand, Delyon, and Mémin [10], and then Cheridito and Stadje [11], studied such stability properties for continuous backward stochastic differential equations (BSDEs for short), a type of non-linear martingale representation. Mémin [52] looked into the stabi...

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“…BSDEs driven by Lévy processes. Notably, only the articles of Bouchard and Élie [11], Aazizi [1] (in the pure jump case), Lejay, Mordecki, and Torres [69], Geiss and Labart [39] (in [39,69] the jump part of the driving martingale is a Poisson process), Kharroubi and Lim [66], with a jump process depending on the Brownian motion itself, Madan, Pistorius, and Stadje [74] (which follows in spirit the approach of [18]) and Dumitrescu and Labart [35] (where the jump part of the driving martingale is a Poisson process and the authors actually consider reflected BSDEs) consider BSDEs which include stochastic integrals with respect to an integer-valued measure, associated to the jumps of a Lévy process. See also Khedher and Vanmaele [67] for BSDEs driven by càdlàg martingales.…”

Section: Introductionmentioning

confidence: 99%

Stability of backward stochastic differential equations: the general case

Papapantoleon¹,

Possamaï²,

Saplaouras³

2021

Preprint

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In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations. CONTENTS 1. INTRODUCTION 1 General notation 4 2. FRAMEWORK 5 3. STABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 8 3.1. Outline of the proof 8 3.2. Uniform a priori BSDE estimates 9 3.3. Reducing the complexity of the induction steps and proving Theorem 3.1 10 3.4. The first step of the induction is valid 15 3.5. The p-th step of the induction is valid 18 3.6. On the nature of the conditions 25 4. EXAMPLES AND APPLICATIONS 26 4.1. Continuous martingales 27 4.2. Probabilistic numerical schemes for deterministic equations 27 4.3. PII martingales 27 4.4. BSDEs as dual problems 28 APPENDIX A. AUXILIARY RESULTS 28 A.1. Definitions and complete notation 28 A.2. Moore-Osgood's theorem 30 A.3. Weak convergence of measures on the positive real line 30 A.4. Some helpful lemmata for the proof of Theorem 3.1. 38 REFERENCES 41

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Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver

Cited by 11 publications

References 27 publications

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

Stability results for martingale representations: The general case

Stability of backward stochastic differential equations: the general case

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