Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T02:31:21.724Z Has data issue: false hasContentIssue false
  • English
  • Français

RECURSIVE BACKWARD SCHEME FOR THE SOLUTION OF A BSDE WITH A NON LIPSCHITZ GENERATOR

Published online by Cambridge University Press:  05 January 2017

Paola Tardelli
Paola Tardelli*
Affiliation:
Department of Industrial and Information Engineering and Economics, University of L'Aquila, Piazzale E. Pontieri 1, 67040, Monteluco di Roio, Italy E-mail: paola.tardelli@univaq.it
Get access Rights & Permissions [Opens in a new window]

Abstract

On an incomplete financial market, the stocks are modeled as pure jump processes subject to defaults. The exponential utility maximization problem is investigated characterizing the value function in term of Backward Stochastic Differential Equations (BSDEs), driven by pure jump processes. In general, in this setting, there is no unique solution. This is the reason why, the value function is proven to be the limit of a sequence of processes. Each of them is the solution of a Lipschitz BSDE and it corresponds to the value function associated with a subset of bounded admissible strategies. Given a representation of the jump processes driving the model, the aim of this note is to give a recursive backward scheme for the value function of the initial problem.

Keywords

backward stochastic differential equationsdefaultincomplete market
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 2 , April 2017 , pp. 207 - 225
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bacry, E., Delattre, S., Hoffmann, M., & Muzy, J.F. (2013). Modeling microstructure noise with mutually exciting point processes. Quantitative Finance 13(1): 65–77.CrossRefGoogle Scholar
2. Barndorff-Nielsen, O.E. & Shephard, N. (2001). Non Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63(2): 167–241.Google Scholar
3. Bielecki, T.R., Jeanblanc, M., & Rutkowski, M. (2006). Hedging of credit derivatives in models with totally unexpected default. In Akahori, J. et al. (eds.), Stochastic Processes and Applications to Mathematical Finance, Proceedings of the 5th Ritsumeikan International Symposium. Singapore: World Scientific Publishing, pp. 35–100.CrossRefGoogle Scholar
4. Bouchard, B. & Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Processes and their Applications 118(1): 53–75.CrossRefGoogle Scholar
5. Bremaud, P. (1981). Point processes and queues. Martingale dynamics. Springer Series in Statistics. New York-Berlin: Springer-Verlag.Google Scholar
6. Carbone, R., Ferrario, B., & Santacroce, M. (2008). Backward stochastic differential equations driven by Cadlag Martingales. Teor. Veroyatn. Primen. Transl. to appear in Theory of Probability and its Applications 52(2): 304–314. DOI:10.1137/S0040585X97983055 CrossRefGoogle Scholar
7. Carr, P., Geman, H., Madan, D.B., & Yor, M. (2002). The fine structure of asset returns: an empirical investigation. Journal of Business 75(2): 305–332.Google Scholar
8. Cartea, A. (2013). Derivatives pricing with marked point processes using Tick-by-Tick data. Quantitative Finance 13(1): 111–123.Google Scholar
9. Ceci, C. (2012). Utility maximization with intermediate consumption under restricted information for jump market models. International Journal of Theoretical and Applied Finance 15: 6, DOI:10.1142/S0219024912500409 Google Scholar
10. Centanni, S. & Minozzo, M. (2006). A Monte Carlo approach to filtering for a class of marked double stochastic Poisson processes. Journal of the American Statistical Association 101(476): 1582–1597.Google Scholar
11. Dellacherie, C. & Meyer, P.A. (1982). Probabilities and Potential. B. Theory of Martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies 72, Amsterdam: North-Holland Publishing.Google Scholar
12. Engle, R.F. & Russell, J.R. (1998). Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66(5): 1127–1162.Google Scholar
13. Ethier, S.N. & Kurtz, T.G. (2005). Markov processes: characterization and convergence. Wiley series in probability and statistics: probability and mathematical statistics. New York: John Wiley Inc. ISBN: 978-0-471-76986-6 Google Scholar
14. Frey, R. & Runggaldier, W.J. (2001). A nonlinear filtering approach to volatility estimation with a view towards high frequency data. International Journal of Theoretical and Applied Finance 4(2): 199. DOI:10.1142/S021902490100095X Google Scholar
15. Gerardi, A. & Tardelli, P. (2006). Filtering a partially observed ultra-high-frequency data model. Acta Applicandae Mathematica 91(2): 193–205.Google Scholar
16. Hu, Y., Imkeller, P., & Muller, M. (2005). Utility maximization in incomplete markets. The Annals of Applied Probability 15(3): 1691–1712.Google Scholar
17. Jeanblanc, M., Yor, M., & Chesney, M. (2009). Mathematical methods for financial markets. Springer Finance, London: Springer-Verlag. ISBN: 978-1-84628-737-4.Google Scholar
18. Jing, B.Y., Kong, X.B., & Liu, Z. (2012). Modeling high-frequency financial data by pure jump processes. The Annals of Statistics 40(2): 759–784.Google Scholar
19. Karatzas, I. & Shreve, S.E. (1998). Brownian motion and stochastic calculus. New York: Springer Verlag. ISBN 978-1-4612-0949-2.Google Scholar
20. Lim, T. & Quenez, M.C. (2011). Exponential utility maximization in an incomplete market with defaults. Electronic Journal of Probability 16(53): 1434–1464.CrossRefGoogle Scholar
21. Madan, D.B., Carr, P.P., & Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review 2: 79–105.Google Scholar
22. Mania, M. & Schweizer, M. (2005). Dynamic exponential utility indifference valuation. The Annals of Applied Probability 15(3): 2113–2143.Google Scholar
23. Mansuy, R. & Yor, M. (2006). Random times and enlargement of filtrations in a Brownian. Lecture Notes in Mathematics 1873. Berlin: Springer-Verlag, ISBN 978-3-540-32416-4.Google Scholar
24. Martin, J.S., Jasra, A., & McCoy, E. (2013). Inference for a class of partially observed point process models. Annals of the Institute of Statistical Mathematics 65(3): 413–437.Google Scholar
25. Pringent, J.L. (2001). Option pricing with a general marked point process. Mathematics of Operations Research 26(1): 50–66.Google Scholar
26. Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. The Annals of Applied Probability 11(3): 694–734.Google Scholar
27. Tardelli, P. (2011). Utility maximization in a pure jump model with partial observation. Probability in the Engineering and Informational Sciences 25(1): 29–54.Google Scholar