Skip to main content
SpringerLink
Log in

On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples

  • Conference paper
  • First Online:
Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications (BSDE-SPDE 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 289))

Included in the following conference series:

Abstract

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure of jumps, which could be of infinite activity with a non-deterministic and time-inhomogeneous compensator. The BSDE generator function can be non-convex and needs not satisfy global Lipschitz conditions in the jump integrand. We contribute concrete sufficient criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori \(L^\infty \)-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Antonelli, F., Mancini, C.: Solutions of BSDEs with jumps and quadratic/locally lipschitz generator. Stochast. Process. Appl. 126(10), 3124–3144 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandini, E.: Existence and uniqueness for backward stochastic differential equations driven by a random measure, possibly non quasi-left continuous. Electron. Commun. Probab. 20, 13 pp. (2015)

    Google Scholar 

  3. Bandini, E., Confortola, F.: Optimal control of semi-markov processes with a backward stochastic differential equations approach. Math. Control Sign. Syst. 29(1), 1 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barles, G., Buckdahn, R., Pardoux, E.: BSDE’s and integral-partial differential equations. Stochastics 60, 57–83 (1997)

    MATH  Google Scholar 

  5. Barrieu, P., El Karoui, N.: Pricing, hedging and designing derivatives with risk measures. In: Carmona, R. (ed.) Indifference Pricing, Theory and Applications, pp. 77–146. Princeton Univ. Press (2009)

    Google Scholar 

  6. Barrieu, P., El Karoui, N.: Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 41(3B), 1831–1863 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Becherer, D.: Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027–2054 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Becherer, D.: Utility indifference valuation. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance. Wiley, Chichester (2010)

    Google Scholar 

  9. Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk 2(1), 13 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Becherer, D., Kentia, K.: Hedging under generalized good-deal bounds and model uncertainty. Math. Meth. Oper. Res. 86(1), 171–214 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Björk, T., Slinko, I.: Towards a general theory of good-deal bounds. Rev. Financ. 10, 221–260 (2006)

    Article  MATH  Google Scholar 

  12. Carr, P., Geman, H., Madan, D., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75(2), 305–332 (2002)

    Article  Google Scholar 

  13. Cochrane, J., Saá Requejo, J.: Good deal asset price bounds in incomplete markets: beyond arbitrage. J. Polit. Econ. 108, 79–119 (2000)

    Google Scholar 

  14. Cohen, S.N., Elliott, R.J.: Stochastic calculus and applications, 2nd edn. In: Probability and Its Applications. Springer, New York (2015)

    Google Scholar 

  15. Cohen, S.N., Elliott, R.J.: Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Probab. 20, 267–311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Confortola, F., Fuhrman, M.: Backward stochastic differential equations associated to jump Markov processes and applications. Stochast. Process. Appl. 124, 289–316 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Confortola, F., Fuhrman, M., Jacod, J.: Backward stochastic differential equations driven by a marked point process: an elementary approach with an application to optimal control. Ann. Appl. Probab. 26(3), 1743–1773 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dellacherie, C., Meyer, P.: Probabilities and potenial B: theory of martingales. In: Mathematics Studies. North Holland, Amsterdam, New York, Oxford (1982)

    Google Scholar 

  19. Delong, L., Pelsser, A.: Instantaneous mean-variance hedging and Sharpe ratio pricing in a regime-switching financial model. Stochast. Models 31, 67–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Di Tella, P., Engelbert, H.-J.: The predictable representation property of compensated-covariation stable families of martingales. Theory Probab. Appl. 60, 99–130 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Di Tella, P., Engelbert, H.-J.: On the predictable representation property of martingales associated with Lévy processes. Stochastics 87(1), 1–15 (2015)

    Google Scholar 

  22. Eddahbi, M., Fakhouri, I., Ouknine, Y.: \(\{L\}^p\) (\(p \ge 2\))-solutions of generalized BSDEs with jumps and monotone generator in a general filtration. Modern Stochast.: Theor. Appl. 4(1), 25–63 (2017)

    MathSciNet  MATH  Google Scholar 

  23. El Karoui, N., Matoussi, A., Ngoupeyou, A.: Quadratic exponential semimartingales and application to BSDEs with jumps. arXiv preprint, arXiv:1603.06191 (2016)

  24. Geiss, C., Steinicke, A.: Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probab. Uncertain. Quant. Risk. 3(1), 9 (2018)

    Google Scholar 

  25. Geiss, C., Steinicke, A.: Malliavin derivative of random functions and applications to Lévy driven BSDEs. Electron. J. Probab. 21, 28 pp. (2016)

    Google Scholar 

  26. Geiss, C., Labart, C.: Simulation of BSDEs with jumps by Wiener Chaos expansion. Stochast. Process. Appl. 126(7), 2123–2162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Geiss, C., Steinicke, A.: \(L_2\)-variation of Lévy driven BSDEs with non-smooth terminal conditions. Bernoulli 22(2), 995–1025 (2016)

    MathSciNet  MATH  Google Scholar 

  28. He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. Science Press, CRC Press, New York (1992)

    MATH  Google Scholar 

  29. Hu, Y., Liang, G., Tang, S.: Exponential utility maximization and indifference valuation with unbounded payoffs. arXiv preprint. arXiv:1707.00199 (2017)

  30. Hu, Y., Imkeller, P., Müller, M.: Utility maximization in incomplete markets. Ann. Appl. Probab. 15, 1691–1712 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  32. Kazamaki, N.: Continuous exponential martingales and BMO. In: Lecture Notes in Mathematics, vol. 1579. Springer, Berlin (1994)

    Google Scholar 

  33. Kazamaki, N.: A sufficient condition for the uniform integrability of exponential martingales. Math. Rep. Toyama University 2, 1–11 (1979)

    MathSciNet  MATH  Google Scholar 

  34. Kazi-Tani, N., Possamaï, D., Zhou, C.: Quadratic BSDEs with jumps: related nonlinear expectations. Stochast. Dyn. 16(4), 1650012, 32 pp. (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kazi-Tani, N., Possamaï, D., Zhou, C.: Quadratic BSDEs with jumps: a fixed-point approach. Electron. J. Probab. 20(66), 1–28 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Kentia, K.: Robust aspects of hedging and valuation in incomplete markets and related backward SDE theory. Ph.D. thesis, Humboldt-Universität zu Berlin. urn:nbn:de:kobv:11-100237580 (2015)

    Google Scholar 

  37. Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Appl. Probab. 28, 558–602 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kruse, T., Popier, A.: BSDEs with monotone generator driven by brownian and poisson noises in a general filtration. Stochastics 88(4), 491–539 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kruse, T., Popier, A.: Lp-solution for BSDEs with jumps in the case \(p<2\). Stochastics 89(8), 1201–1227 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Laeven, R.J.A., Stadje, M.A.: Robust portfolio choice and indifference valuation. Math. Oper. Res. 39, 1109–1141 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lepingle, D., Mémin, J.: Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrscheinlichkeitstheor. verw. Geb. 42, 175–203 (1978)

    Article  MATH  Google Scholar 

  42. Mania, M., Chikvinidze, B.: New proofs of some results on BMO martingales using BSDEs. J. Theoret. Probab. 27, 1213–1228 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15(3), 2113–2143 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Morlais, M.: Utility maximization in a jump market model. Stochastics 81, 1–27 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Morlais, M.: A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem. Stochast. Process. Appl. 120, 1966–1995 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stochast. Process. Appl. 90, 109–122 (2000)

    Article  MATH  Google Scholar 

  47. Nualart, D., Schoutens, W.: Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance. Bernoulli 7(5), 761–776 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nutz, M.: The opportunity process for optimal consumption and investment with power utility. Math. Financ. Econ. 3(3–4), 139–159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Papapantoleon, A., Possamaï, D., Saplaouras, A.: Existence and uniqueness results for BSDEs with jumps: the whole nine yards. Electron. J. Probab. 23, Paper no. 121, 68 pp (2018)

    Google Scholar 

  50. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Protter, P., Shimbo, K.: No arbitrage and general semimartingales. Markov Processes and related Topics: A Festschrift for Thomas G. Kurtz 4, 267–283 (2008)

    Google Scholar 

  52. Quenez, M.-C., Sulem, A.: BSDEs with jumps, optimization and applications to dynamic risk measures. Stochast. Process. Appl. 123(8), 3328–3357 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Waelbroeck, L. (ed.) Nonlinear Operators and Calculus of Variations. Lecture Notes in Mathematics 543, pp. 157–207. Springer, Berlin (1976)

    Chapter  Google Scholar 

  54. Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stochast. Process. Appl. 116, 1358–1376 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tang, S., Li, X.: Necessary conditions for optimal control for stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447–1475 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Tevzadze, R.: Solvability of backward stochastic differential equations with quadratic growth. Stochast. Process. Appl. 118(3), 503–515 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yao, S.: Lp solutions of backward stochastic differential equations with jumps. Stochast. Process. Appl. 127(11), 3465–3511 (2017)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Dirk Becherer

  2. Hochschule für Technik und Wirtschaft, Treskowallee 8, 10313, Berlin, Germany

    Martin Büttner

  3. Institut für Mathematik, Goethe-Universität Frankfurt, 60054, Frankfurt am Main, Germany

    Klebert Kentia

Authors
  1. Dirk Becherer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Büttner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klebert Kentia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dirk Becherer .

Editor information

Editors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Samuel N. Cohen

  2. School of Mathematics, University of Edinburgh, Edinburgh, UK

    István Gyöngy

  3. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Gonҫalo dos Reis

  4. School of Mathematics, University of Edinburgh, Edinburgh, UK

    David Siska

  5. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Łukasz Szpruch

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Becherer, D., Büttner, M., Kentia, K. (2019). On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples. In: Cohen, S., Gyöngy, I., dos Reis, G., Siska, D., Szpruch, Ł. (eds) Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications. BSDE-SPDE 2017. Springer Proceedings in Mathematics & Statistics, vol 289. Springer, Cham. https://doi.org/10.1007/978-3-030-22285-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation