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On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

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  • Dirk Becherer
  • Martin Buttner
  • Klebert Kentia

Abstract

We extend the monotone stability approach for backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure, which can be of infinite activity and time-inhomogeneous with non-deterministic compensator. The BSDE generator function can be non-convex and needs not to satisfy classical global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, and extended results for comparison and for existence and uniqueness of bounded solutions to BSDEs with jumps. The scope of results, applicability of assumptions and differences to related results by some alternative approaches are demonstrated by several examples for control problems from finance.

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  • Dirk Becherer & Martin Buttner & Klebert Kentia, 2016. "On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples," Papers 1607.06644, arXiv.org.
  • Handle: RePEc:arx:papers:1607.06644
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    References listed on IDEAS

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    1. Dirk Becherer & Martin Schweizer, 2005. "Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes," Papers math/0505208, arXiv.org.
    2. Confortola, Fulvia & Fuhrman, Marco, 2014. "Backward stochastic differential equations associated to jump Markov processes and applications," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 289-316.
    3. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    4. Samuel N. Cohen & Robert J. Elliott, 2008. "Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions," Papers 0810.0055, arXiv.org, revised Jan 2010.
    5. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    6. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    7. Tevzadze, Revaz, 2008. "Solvability of backward stochastic differential equations with quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 503-515, March.
    8. Marcel Nutz, 2009. "The Bellman equation for power utility maximization with semimartingales," Papers 0912.1883, arXiv.org, revised Mar 2012.
    9. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    10. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    11. Geiss, Christel & Labart, Céline, 2016. "Simulation of BSDEs with jumps by Wiener Chaos expansion," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2123-2162.
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