Reflected backward doubly stochastic differential equations with discontinuous barrier

Berrhazi, Badr-eddine; Fatini, Mohamed El; Hilbert, Astrid; Mrhardy, Naoual; Pettersson, Roger

doi:10.1080/17442508.2019.1691207

Cited by 7 publications

(5 citation statements)

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“…where d ← B r is a so-called backward Itô integral. We refer to, e.g., [3,4,5,6,8,11,36,39,42] for more discussion on RBDSDEs, which plays an important role in SPDEs, infinite dimensional stochastic control and nonlinear filtering. Our research on RBSDE with a rough driver can provide a new perspective for the study of RBDSDE.…”

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confidence: 99%

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

Péng

Hima

2017

Sci. China Math.

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In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.The solution of this equation consists of a triplet of processes (Y, Z, K). The existence and uniqueness of the solution are proved in [10]. In [11] the comparison theorem, Feymann-Kac formula and some related topics associated with this kind of G-BSDEs were established.In this paper, we study the case where the solution of a G-BSDE is required to stay above a given stochastic process, called the lower obstacle. An increasing process should be added in this equation to push the solution upwards, so that it may remain above the obstacle. According to the classical case studied by [5], we may expect that the solution of reflected G-BSDE is a quadrupleThe shortcoming is that the solution of the above problem is not unique. Thus, to get the uniqueness of the reflected G-BSDE, we should reformulate this problem as the following. A triplet of processes (Y, Z, A) is called a solution of reflected G-BSDE if the following properties hold:is a non-increasing G-martingale. Here, we denote by S α G (0, T ) the collection of process (Y, Z, A) such that Y ∈ S α G (0, T ), Z ∈ H α G (0, T ), A is a continuous nondecreasing process with A 0 = 0 and A ∈ S α G (0, T ). Note that we use a "martingale condition" (c) instead of the Skorohod condition. Under some appropriate assumptions, we can prove that the solution of the above reflected G-BSDE is unique. In proving the existence of this problem, we should use the approximation method via penalization. This is a constructive method in the sense that the solution of the reflected G-BSDE is proved to be the limit of a sequence of penalized G-BSDEs. Different from the classical case, the dominated convergence theorem does not hold under G-framework. Besides, any bounded sequence in M p G (0, T ) is no longer weakly compact. The main difficulty in carrying out this construction is to prove the convergence property in some appropriate sense. It is worth pointing out that the main idea is to apply the uniformly continuous property of the elements in S p G (0, T ). Actually, the above equations hold P -a.s. for every probability measure P belongs to a nondominated class of mutually singular measures. Therefore, the G-expectation theory shares many similarities with second order BSDEs (2BSDEs for short) developed by Cheridito, Soner, Touzi and Victoir [1]. Matoussi, Possamai and Zhou [21] showed the existence and uniqueness of second order reflected BSDE whose solution is (Y, Z, K P ) P ∈P κ H satisfyingwith Y t ≥ S t , K P t − k P t = ess inf P P ′ ∈PH (t+,P ) E P ′ t [K P T − k P T ], P -a.s., 0 ≤ t ≤ T, ∀P ∈ P κ H ,

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confidence: 99%

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

Péng

Hima

2017

Sci. China Math.

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“…With the wide applications into some important fields such as mechanics, communication, aerospace engineering, economic management and financial market, many mathematical models characterized by stochastic differential equations (SDEs) have been proposed in [18,28,29] and the references therein. In recent years, stability analysis, the derivation principle, the ergodicity, and the attracting for SDEs have been investigated in [21,24,13,1]. Among these properties, stability analysis is an important and fundamental one, and some valuable methodologies as well as results have been presented in [2,9,12,7,27], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

Chang

Chen

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th(<inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>) for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M4">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.</p>

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“…In all of the mentioned works, the barrier has been assumed to be at least right continuous. Recently, Berrhazi et al [8] considered RBDSDEs when the barrier is not necessarily right-continuous by inspiring on the work of Grigorova et al [15] which is the first one dealing with right upper semicontinuous barrier reflected BSDEs. For more developments on Reflected BSDEs when the barrier is not necessarily right-continuous, we refer to [1,6,16,19,23,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…For more developments on Reflected BSDEs when the barrier is not necessarily right-continuous, we refer to [1,6,16,19,23,25,26]. More recently, Marzougue and Sagna [27] extended the work of Berrhazi et al [8] to the case when the noise is driven also by an independent Poisson random measure under the so-called stochastic Lipschitz condition on the drivers.…”

Section: Introductionmentioning

confidence: 99%

Penalization method for reflected BDSDEs with two-sided jumps and driven by Lévy process

Marzougue¹

2021

Preprint

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Reflected backward doubly stochastic differential equations with discontinuous barrier

Cited by 7 publications

References 25 publications

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

Penalization method for reflected BDSDEs with two-sided jumps and driven by Lévy process

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