Partially informed investors: hedging in an incomplete market with default. (English) Zbl 1345.49045

Tardelli, P.

J. Appl. Probab. 52, No. 3, 718-735 (2015).

Summary: 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.

MSC:

49N30	Problems with incomplete information (optimization)
60H30	Applications of stochastic analysis (to PDEs, etc.)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
49J55	Existence of optimal solutions to problems involving randomness
49L20	Dynamic programming in optimal control and differential games
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
93E20	Optimal stochastic control
91G80	Financial applications of other theories
93E11	Filtering in stochastic control theory
93E03	Stochastic systems in control theory (general)
90C39	Dynamic programming

Keywords:

optimal investment; incomplete market; partial information; exponential utility; default time; backward stochastic differential equation; dynamic programming; marked point process; filtering

References:

[1]	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 , J. Akahori \et (eds), World Scientific, Singapore, 35-100 · Zbl 1186.91190
[2]	Bouchard, B. and Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Process. Appl. 118, 53-75. · Zbl 1136.60048
[3]	Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics . Springer, New York.
[4]	Carbone, R., Ferrario, B. and Santacroce, M. (2008). Backward stochastic differential equations driven by càdlàg martingales. Theory Prob. Appl. 52, 304-314. · Zbl 1152.60050
[5]	Ceci, C. (2006). Risk minimizing hedging for a partially observed high frequency data model. Stochastics 78, 13-31. · Zbl 1156.91362
[6]	Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales . North-Holland, Amsterdam. · Zbl 0494.60002
[7]	Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence . John Wiley, New York. · Zbl 0592.60049
[8]	Frey, R. (2000). Risk minimization with incomplete information in a model for high-frequency data. Math. Finance 10, 215-225. · Zbl 1022.91023
[9]	Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10, 39-52. · Zbl 1013.60026
[10]	Gerardi, A. and Tardelli, P. (2010). Stochastic control methods: hedging in a market described by pure jump processes. Acta Appl. Math. 111, 233-255. · Zbl 1206.60076
[11]	Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Prob. 15, 1691-1712. · Zbl 1083.60048
[12]	Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714 ). Springer, Berlin. · Zbl 0414.60053
[13]	Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets . Springer, London. · Zbl 1205.91003
[14]	Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0638.60065
[15]	Lim, T. and Quenez, M.-C. (2011). Exponential utility maximization in an incomplete market with defaults. Electron. J. Prob. 16, 1434-1464. · Zbl 1245.49039
[16]	Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Prob. 15, 2113-2143. · Zbl 1134.91449
[17]	Prigent, J.-L. (2001). Option pricing with a general marked point process. Math. Operat. Res. 26, 50-66. · Zbl 1073.91592
[18]	Runggaldier, W. J. (2003). Jump-diffusion models. In Handbook of Heavy Tailed Distributions in Finance , Elsevier, Amsterdam, pp. 169-209.
[19]	Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Prob. 11, 694-734. · Zbl 1049.91085
[20]	Tardelli, P. (2011). Utility maximization in a pure jump model with partial observation. Prob. Eng. Inf. Sci. 25, 29-54. · Zbl 1209.91188
[21]	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.
[22]	Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61-82. · Zbl 0977.93081