Home Business & Economics BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem
Article
Licensed
Unlicensed Requires Authentication

BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

  • Khalid Oufdil ORCID logo EMAIL logo
Published/Copyright: April 29, 2021
Become an author with De Gruyter Brill

Abstract

In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable (|z||ln|z||). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

MSC 2010: 93E20; 60H10; 60H99

Award Identifier / Grant number: 18UIZ2017

Funding statement: This work has been done under the scholarship of the National Center for Scientific and Technical Research, Morocco.

References

[1] K. Bahlali, Backward stochastic differential equations with locally Lipschitz coefficient, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 5, 481–486. 10.1016/S0764-4442(01)02063-8Search in Google Scholar

[2] K. Bahlali, Existence, uniqueness and stability for solutions of backward stochastic differential equations with locally Lipschitz coefficient, Electron. Commun. Probab. 7 (2002), 169–179. 10.1214/ECP.v7-1058Search in Google Scholar

[3] K. Bahlali and B. El Asri, Stochastic optimal control and BSDEs with logarithmic growth, Bull. Sci. Math. 136 (2012), no. 6, 617–637. 10.1016/j.bulsci.2011.12.008Search in Google Scholar

[4] K. Bahlali, E. Essaky and M. Hassani, Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs, C. R. Math. Acad. Sci. Paris 348 (2010), no. 11–12, 677–682. 10.1016/j.crma.2010.03.006Search in Google Scholar

[5] K. Bahlali, E. H. Essaky, M. Hassani and E. Pardoux, Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient, C. R. Math. Acad. Sci. Paris 335 (2002), no. 9, 757–762. 10.1016/S1631-073X(02)02542-6Search in Google Scholar

[6] K. Bahlali, O. Kebiri, N. Khelfallah and H. Moussaoui, One dimensional BSDEs with logarithmic growth application to PDEs, Stochastics 89 (2017), no. 6–7, 1061–1081. 10.1080/17442508.2017.1311900Search in Google Scholar

[7] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep. 60 (1997), no. 1–2, 57–83. 10.1080/17442509708834099Search in Google Scholar

[8] V. E. Beneš, Existence of optimal stochastic control laws, SIAM J. Control 9 (1971), 446–472. 10.1137/0309034Search in Google Scholar

[9] A. Bensoussan and J.-L. Lions, Contrôle impulsionnel et inéquations quasi variationnelles, Dunod, Paris, 1982. Search in Google Scholar

[10] T. Kruse and A. Popier, BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 88 (2016), no. 4, 491–539. 10.1080/17442508.2015.1090990Search in Google Scholar

[11] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Universitext, Springer, Berlin, 2005. Search in Google Scholar

[12] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. 10.1016/0167-6911(90)90082-6Search in Google Scholar

Received: 2021-04-03
Accepted: 2021-04-14
Published Online: 2021-04-29
Published in Print: 2021-06-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 6.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/eqc-2021-0012/pdf
Scroll to top button