Backward stochastic differential equations with stochastic monotone coefficients

Bahlali, Khaled; Elouaflin, Abouo; N'zi, Modeste

doi:10.1155/s1048953304310038

Cited by 15 publications

(9 citation statements)

References 7 publications

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“…Define P loc the subset of M 1 such that, for each P ∈ P loc , X is P-local martingale whose quadratic variation X is absolutely continuous in t with respect to the Lebesgue measure. Note that the d × d-matrix-valued processes X can be defined pathwisely, and we may introduce the corresponding F-progressively measurable density processes a t := lim sup n→∞ n X t − X t− 1 n , EJP 25 (2020), paper 99.…”

Section: Canonical Spacementioning

confidence: 99%

“…. < t n = T of [0, T ], where the representation follows from a backward resolution of the heat equation ∂ t v + 1 2 ∆v = 0 on each time interval [t i−1 , t i ], i = 1, . .…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Canonical Spacementioning

confidence: 99%

“…. < t n = T of [0, T ], where the representation follows from a backward resolution of the heat equation ∂ t v + 1 2 ∆v = 0 on each time interval [t i−1 , t i ], i = 1, . .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second order backward SDE with random terminal time

Lin¹,

Ren²,

Touzi³

et al. 2020

Electron. J. Probab.

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“…where f 1 , f 2 are d−dimensional G−optional processes such that 9 In the proof of [134,Proposition 25.4] we can find a convenient tool for constructing sub-multiplicative functions.…”

Section: )mentioning

confidence: 99%

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

Papapantoleon¹,

Possamaï²,

Saplaouras³

2018

Electron. J. Probab.

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This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete-time approximations of general martingales.2010 Mathematics Subject Classification. 60G48, 60G55, 60G57, 60H05. Key words and phrases. BSDEs, processes with jumps, stochastically discontinuous martingales, random time horizon, stochastic Lipschitz generator. We thank Martin Schweizer, two anonymous referees and the associate editor for their comments that have resulted in a significant improvement of the manuscript. Alexandros Saplaouras gratefully acknowledges the financial support from the DFG Research Training Group 1845 "Stochastic Analysis with Applications in Biology, Finance and Physics". Dylan Possamaï gratefully acknowledges the financial support of the ANR project PACMAN, ANR-16-CE05-0027. Moreover, all authors gratefully acknowledge the financial support from the PROCOPE project "Financial markets in transition: mathematical models and challenges". 1 The authors are indebted to Saïd Hamadène for pointing out this reference. The published version of [51] states that the article was received on October 27, 1971. It is also present in the bibliography of [21], though it is never referred to in the text. 2 We emphasize that the references given below are just the tip of the iceberg, though most of them are, in our view, among the major ones of the field. Nonetheless, we do not make any claim about comprehensiveness of the following list. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS mention El Karoui and Huang [60], Bender and Kohlmann [18], Wang, Ran and Chen [138] as well as Briand and Confortola [27]. The first results going beyond the linear growth assumption in z, which assumed quadratic growth, were obtained independently by Kobylanski [97, 98, 99] and Dermoune, Hamadène and Ouknine [56], for bounded ξ and f Lipschitz in y. These results were then further studied by Eddhabi and Ouknine [59], and improved by Lepeltier and San Martín [107, 108], Briand, Lepeltier and San Martín [35] and revisited by Briand and Élie [31], but still for bounded ξ. Wellposedness in the quadratic case when ξ has sufficiently large exponential moments was then investigated by Briand and Hu [33, 34], followed by Delbaen, Hu and Richou [54, 55], Essaky and Hassani [66], and Briand and Richou [36]. A specific quadratic setting with only square integrable terminal conditions has been considered recently by Bahlali, Eddahbi and Ouknine [6, 7], while a result with logarithmic growth was also obtained by Bahlali and El Asri [8], and Bahlali, Kebiri, Khelfallah and Moussaoui [13]. The ca...

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“…To this direction, several attempts have been done. Among others, we refer to [4,5,9,15,[21][22][23][24] for the case of BSDEs, and [16,25,26,30] for BDSDEs. In our paper, we use a generalization of the Doob-Meyer decomposition called the Mertens decomposition.…”

Section: Introductionmentioning

confidence: 99%

Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions

Marzougue¹,

Sagna²

2020

Modern Stochastics: Theory and Applications

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In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.

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Backward stochastic differential equations with stochastic monotone coefficients

Cited by 15 publications

References 7 publications

Second order backward SDE with random terminal time

Second order backward SDE with random terminal time

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

Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions

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