Skip to main content Accessibility help
×
Home

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

  • Paola Tardelli (a1)

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.

Copyright

References

Hide All
1. Bacry, E., Delattre, S., Hoffmann, M., & Muzy, J.F. (2013). Modeling microstructure noise with mutually exciting point processes. Quantitative Finance 13(1): 6577.
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): 167241.
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. 35100.
4. Bouchard, B. & Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Processes and their Applications 118(1): 5375.
5. Bremaud, P. (1981). Point processes and queues. Martingale dynamics. Springer Series in Statistics. New York-Berlin: Springer-Verlag.
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): 304314. DOI:10.1137/S0040585X97983055
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): 305332.
8. Cartea, A. (2013). Derivatives pricing with marked point processes using Tick-by-Tick data. Quantitative Finance 13(1): 111123.
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
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): 15821597.
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.
12. Engle, R.F. & Russell, J.R. (1998). Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66(5): 11271162.
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
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
15. Gerardi, A. & Tardelli, P. (2006). Filtering a partially observed ultra-high-frequency data model. Acta Applicandae Mathematica 91(2): 193205.
16. Hu, Y., Imkeller, P., & Muller, M. (2005). Utility maximization in incomplete markets. The Annals of Applied Probability 15(3): 16911712.
17. Jeanblanc, M., Yor, M., & Chesney, M. (2009). Mathematical methods for financial markets. Springer Finance, London: Springer-Verlag. ISBN: 978-1-84628-737-4.
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): 759784.
19. Karatzas, I. & Shreve, S.E. (1998). Brownian motion and stochastic calculus. New York: Springer Verlag. ISBN 978-1-4612-0949-2.
20. Lim, T. & Quenez, M.C. (2011). Exponential utility maximization in an incomplete market with defaults. Electronic Journal of Probability 16(53): 14341464.
21. Madan, D.B., Carr, P.P., & Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review 2: 79105.
22. Mania, M. & Schweizer, M. (2005). Dynamic exponential utility indifference valuation. The Annals of Applied Probability 15(3): 21132143.
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.
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): 413437.
25. Pringent, J.L. (2001). Option pricing with a general marked point process. Mathematics of Operations Research 26(1): 5066.
26. Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. The Annals of Applied Probability 11(3): 694734.
27. Tardelli, P. (2011). Utility maximization in a pure jump model with partial observation. Probability in the Engineering and Informational Sciences 25(1): 2954.

Keywords

Related content

Powered by UNSILO

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

  • Paola Tardelli (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.