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Partially informed investors: hedging in an incomplete market with default

Part of: Hamilton-Jacobi theories, including dynamic programming Stochastic systems and control

Published online by Cambridge University Press:  30 March 2016

P. Tardelli
P. Tardelli*
Affiliation:
University of L'Aquila
*
∗ Postal address: Department of Industrial and Information Engineering and Economics, University of L'Aquila, 67100, Italy. Email address: [email protected]
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Abstract

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In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

Keywords

Optimal investmentexponential utilitydefault timedynamic programmingbackward stochastic differential equationfiltering

MSC classification

Primary: 49L20: Dynamic programming method
Secondary: 93E11: Filtering 93E03: Stochastic systems, general
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 718 - 735
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2006). Hedging of credit derivatives in models with totally unexpected default. In Stochastic Processes and Applications to Mathematical Finance, Akahori, J. et al. (eds), World Scientific, Singapore, 35–100 Google Scholar
Bouchard, B. and Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Process. Appl. 118 53-75.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.Google Scholar
Carbone, R., Ferrario, B. and Santacroce, M. (2008). Backward stochastic differential equations driven by càdlàg martingales. Theory Prob. Appl. 52 304-314.Google Scholar
Ceci, C. (2006). Risk minimizing hedging for a partially observed high frequency data model. Stochastics 78 13-31.Google Scholar
Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland, Amsterdam.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Frey, R. (2000). Risk minimization with incomplete information in a model for high-frequency data. Math. Finance 10 215-225.Google Scholar
Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39-52.Google Scholar
Gerardi, A. and Tardelli, P. (2010). Stochastic control methods: hedging in a market described by pure jump processes. Acta Appl. Math. 111 233-255.Google Scholar
Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Prob. 15 1691-1712.Google Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714), Springer, Berlin.Google Scholar
Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer, London.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Lim, T. and Quenez, M.-C. (2011). Exponential utility maximization in an incomplete market with defaults. Electron. J. Prob. 16 1434-1464.Google Scholar
Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Prob. 15 2113-2143.Google Scholar
Prigent, J.-L. (2001). Option pricing with a general marked point process. Math. Operat. Res. 26 50-66.Google Scholar
Runggaldier, W. J. (2003). Jump-diffusion models. In Handbook of Heavy Tailed Distributions in Finance, Elsevier, Amsterdam, pp. 169–209.Google Scholar
Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Prob. 11 694-734.CrossRefGoogle Scholar
Tardelli, P. (2011). Utility maximization in a pure jump model with partial observation. Prob. Eng. Inf. Sci. 25 29-54.Google Scholar
Tardelli, P. (2012). Modeling an incomplete market with default. In Proceedings of the 2nd International Conference on Stochastic Modelling and Simulation, Hikey Media, Royapettah, pp. 54–65. ISBN: 978-81-925286-4-9.Google Scholar
Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5 61-82.Google Scholar