Existence and uniqueness results for BSDE with jumps: the whole nine yards

Papapantoleon, Antonis; Possamaï, Dylan; Saplaouras, Alexandros

doi:10.1214/18-ejp240

Cited by 33 publications

(89 citation statements)

References 130 publications

(217 reference statements)

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“…A series of works [16,30,32,15,18,56,19] are dedicated to the theory of BSDEs (1.3) but driven by a càdlàg martingale under a right-continuous filtration that is also quasi-left continuous. Lately, [12,62] removed the quasileft continuity assumption from the filtration so that the quadratic variation of the driving martingale does not need to be absolutely continuous. On the other hand, based on a general martingale representation result due to Davis and Varaiya [28], Cohen and Elliott [22,23] discussed the case where the driving martingales are not a priori chosen but imposed by the filtration; see Hassani and Ouknine [38] for a similar approach on a BSDE in form of a generic map from a space of semimartingales to the spaces of martingales and those of finite-variation processes.…”

Section: Introductionmentioning

confidence: 99%

Lp Solutions of Backward Stochastic Differential Equations with Jumps

Yao

2016

SSRN Journal

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Given p ∈ (1, 2), we study L p solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in (y, z)−variables. We show that such a BSDEJ with p−integrable terminal data admits a unique L p solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.Keywords: Backward stochastic differential equations with jumps, L p solutions, monotonic generators, convolution with mollifiers. over a probability space (Ω, F , P ) on which B is a Brownian motion and p is an X −valued Poisson point process independent of B. Practically speaking, if the Brownian motion stands for the noise from the financial market, then the Poisson random measure can be interpreted as the randomness of insurance claims. In the BSDEJ (1.1) with generator f and terminal data ξ, a solution consists of an adapted càdlàg process Y , a locally square-integrable predictable process Z and a locally p−integrable predictable random field U .The backward stochastic equation (BSDE) was introduced by Bismut [11] as the adjoint equation for the Pontryagin maximum principle in stochastic control theory. Later, Pardoux and Peng [64] commenced a systematical research of BSDEs. Since then, the BSDE theory has grown rapidly and has been applied to various areas such as mathematical finance, theoretical economics, stochastic control and optimization, partial differential equations, differential geometry and etc, (see the references in [32,27]).Li and Tang [70] introduced into the BSDE a jump term that is driven by a Poisson random measure independent of the Brownian motion. These authors obtained the existence of a unique solution to a BSDEJ with a Lipschitz generator and square-integrable terminal data. Then Barles, Buckdahn and Pardoux [17,7] showed that the wellposedness of BSDEJs gives rise to a viscosity solution of a semilinear parabolic partial integro-differential equation (PIDE) and thus provides a probabilistic interpretation of such a PIDE. Later, Pardoux [63] relaxed the Lipschitz condition of the generator on variable y by assuming a monotonicity condition on variable y instead. Situ [69] and Mao and Yin [76] even degenerated the monotonicity condition of the generator to a weaker version so as to remove the Lipschitz condition on variable z.During the development of the BSDE theory, some efforts were made in relaxing the square integrability on the terminal data so as to be compatible with the fact that linear BSDEs are well-posed for integrable terminal data or that linear expectations have L 1 domains: El Karoui et al. [32] showed that for any p−integrable terminal data, the BSDE with a Lipschitz generator admits a unique L p −solution. Then Briand and Carmona [13] reduced the Lipschitz condition of the generator on variable y by a strong monotonicity condition as well as a polynomial growth

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Section: Introductionmentioning

confidence: 99%

Lp Solutions of Backward Stochastic Differential Equations with Jumps

Yao

2016

SSRN Journal

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“…For µ = 0, i.e., the generator f is monotone, Royer [38] provided the existence and uniqueness under assumptions that the generator f depending only on z is bounded and ξ is bounded. This result was later generalized by Hu & Tessitore [20], Briand & Confortola [6] and Papapantoleon et al [30] to a more general setting. Our Theorem 3.4 generalizes these previous results by allowing for µ ≤ 0, thanks to the new norms under which we set up the wellposedness result.…”

Section: Random Horizon Backward Sdementioning

confidence: 79%

“…On the other hand, they give probabilistic interpretation to solutions of semilinear elliptic PDEs. As our interest in this paper is on the random horizon setting, we refer the interested reader to the related works by El Karoui & Huang [13], Briand & Hu [7], Briand & Carmona [5], Bender & Kohlmann [2], Royer [38], Bahlali, Elouaflin & N'zi [1], Hu and Tessitore [20], Popier [33], Briand and Confortola [6], Wang, Ran and Chen [43], Papapantoleon, Possamaï and Saplaouras [30]. We also mention the related works of Hamadène, Lepeltier & Wu [16], Chen & Wang [8] and Hu and Schweizer [19], which study BSDEs with infinite horizon.…”

Section: Introductionmentioning

confidence: 99%

Second order backward SDE with random terminal time

Lin¹,

Ren²,

Touzi³

et al. 2020

Electron. J. Probab.

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Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).

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“…The following results will allow us to obtain the orthogonal decomposition as understood in Definition 3.5. Their proofs can be found in [59,Appendix A]. For the statement of the following results we will fix an arbitrary filtration F.…”

Section: 22mentioning

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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Existence and uniqueness results for BSDE with jumps: the whole nine yards

Cited by 33 publications

References 130 publications

L<sup><i>p</i></sup> Solutions of Backward Stochastic Differential Equations with Jumps

L<sup><i>p</i></sup> Solutions of Backward Stochastic Differential Equations with Jumps

Second order backward SDE with random terminal time

Stability results for martingale representations: The general case

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