Classical solutions to reaction-diffusion systems for hedging problems with interacting ITÔ and point processes

Annals of Applied Probability

Volumn 15, Issue 2, 2005, Pages 1111-1144

  Becherer, Dirk ^a   Schweizer, Martin ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b ETH ZURICH   (Switzerland)

Author keywords

Credit risk; Hedging; Interacting processes; Reaction diffusion systems; Recursive valuation; Risk minimization

Indexed keywords

EID: 21244505106     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051604000000846     Document Type: Article

References (36)

1. ARVANITIS, A. and GREGORY, J. (2001). Credit: The Complete Guide to Pricing, Hedging and Risk Management. Risk Books, London.
2. BECHERER, D. (2001). Rational hedging and valuation with utility-based preferences. Ph.D. thesis, TU Berlin. Available at edocs.tu-berlin.de/diss/2001/ becherer_dirk.htm.
3. BECHERER, D. (2004). Utility-indifference hedging and valuation via reaction-diffusion systems. Proc. Roy. Soc. London Ser. A 460 27-51.
4. BÉLANGER, A., SHREVE, S. and WONG, D. (2004). A general framework for pricing credit risk. Math. Finance 14 317-350.
5. BIELECKI, T. and RUTKOWSKI, M. (2002). Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin.
6. BLANCHET-SCALLIET, C. and JEANBLANC, M. (2004). Hazard rate for credit risk and hedging defaultable contingent claims. Finance and Stochastics 8 145-159.
7. BRÉMAUD, P. (1981). Point Processes and Queues. Springer, Berlin.
8. DAVIS, M. and LISCHKA, F. (2002). Convertible bonds with market risk and credit risk. In Applied Probability. Studies in Advanced Mathematics (D. Chan, Y.-K. Kwok, D. Yao and Q. Zhang, eds.) 45-58. Amer. Math. Soc., Providence, RI.
9. DAVIS, M. and LO, V. (2001). Modelling default correlation in bond portfolios. In Mastering Risk 2. Applications (C. Alexander, ed.) 141-151. Financial Times/Prentice Hall, Englewood Cliffs, NJ.
10. DI MASI, G., KABANOV, Y. and RUNGGALDIER, W. (1994). Mean-variance hedging of options on stocks with Markov volatilities. Theory Probab. Appl. 39 172-182.
11. DUFFIE, D. and EPSTEIN, L. (1992). Stochastic differential utility. Econometrica 60 353-394.
12. DUFFIE, D., SCHRODER, M. and SKIADAS, C. (1996). Recursive valuation of defaultable securities and the timing of resolution of uncertainty. Ann. Appl. Probab. 6 1075-1090.
13. FÖLLMER, H. and SCHWEIZER, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis. Stochastics Monographs 5 (M. H. A. Davis and R. J. Elliott, eds.) 389-414. Gordon and Breach, New York.
14. FÖLLMER, H. and SONDERMANN, D. (1986). Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics in Honor of Gerard Debreu (W. Hildenbrand and A. Mas-Colell, eds.) 205-223. North-Holland, Amsterdam.
15. FREIDLIN, M. (1985). Functional Integration and Partial Differential Equations. Princeton, New Jersey.
16. FRIEDMAN, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York.
17. HEATH, D. and SCHWEIZER, M. (2001). Martingales versus PDEs in finance: An equivalence result with examples. J. Appl. Probab. 37 947-957.
18. JACOD, J. and PROTTER, P. (1982). Quelques remarques sur un nouveau type d'équations différentielles stochastiques. Séminaire de Probabilités XVI. Lecture Notes in Math. 920 447-458. Springer, Berlin.
19. JARROW, R. and YU, F. (2001). Counterparty risk and the pricing of derivative securities. J. Finance 56 1765-1799.
20. JEANBLANC, M. and RUTKOWSKI, M. (2003). Modelling and hedging of credit risk. In Credit Derivatives: The Definitive Guide (J. Gregory, ed.) 385-416. Risk Books, London.
21. KRYLOV, N. (1987). Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel Publishing and Kluver Academic Publishers, Dordrecht.
22. KUNITA, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. École d'Été de Probabilités de Saint-Flour XII. Lecture Notes in Math. 1097 142-303. Springer, Berlin.
23. KUSUOKA, S. (1999). A remark on default risk models. Advances in Mathematical Economics 1 69-82.
24. LANDO, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2 99-120.
25. MADAN, D. and UNAL, H. (1998). Pricing the risk of default. Review of Derivatives Research 2 121-160.
26. MØLLER, T. (2001). Risk-minimizing hedging strategies for insurance payment processes. Finance and Stochastics 5 419-446.
27. MØLLER, T. (2003). Indifference pricing of insurance contracts in a product space model. Finance and Stochastics 7 197-217.
28. PARDOUX, E. (1999). BSDEs, weak convergence and homogenization of semilinear PDEs. In Nonlinear Analysis, Differential Equations and Control (F. H. Clarke and R. J. Stern, eds.) 503-549. Kluwer Academic, Dordrecht.
29. PAZY, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin.
30. PROTTER, P. (2004). Stochastic Integration and Differential Equations. Springer, Berlin.
31. RAMLAU-HANSEN, H. (1990). Thiele's differential equation as a tool in product development in life insurance. Scand. Actuar. J. 97-104.
32. SCHÖNBUCHER, P. J. (2003). Credit Derivatives Pricing Models. Wiley, Chichester.
33. SCHWEIZER, M. (1991). Option hedging for semimartingales. Stochastic Process. Appl. 37 339-363.
34. SCHWEIZER, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management (E. Jouini, J. Cvitanić and M. Musiela, eds.) 538-574. Cambridge Univ. Press.
35. SMOLLER, J. (1994). Shock Waves and Reaction-Diffusion Equations. Springer, Berlin.
36. STEFFENSEN, M. (2000). A no arbitrage approach to Thiele's differential equation. Insurance Math. Econom. 27 201-214.