Optimal portfolio and consumption selection with default risk

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• 作者：Lijun Bo (1)
  Yongjin Wang (2)
  Xuewei Yang (3)
  
• 关键词：Defaultable security ; average past consumption ; Hamilton ; Jacobi ; Bellman (HJB) equation ; post(pre) ; default ; constrained viscosity solution ; 60H30
• 刊名：Frontiers of Mathematics in China
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：7
• 期：6
• 页码：1019-1042
• 全文大小：229KB
• 参考文献：1. Atar R, Budhiraja A, Williams R J. HJB equations for certain singularly controlled diffusions. Ann Appl Probab, 2007, 17(5鈥?): 1745鈥?776 CrossRef
  2. B茅langer A, Shreve S, Wong D. A general framework for pricing credit risk. Math Finance, 2004, 14(3): 317鈥?50 CrossRef
  3. Benth F E, Karlsen K H, Reikvam K. Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach. Finance Stoch, 2001, 5: 275鈥?03 CrossRef
  4. Biagini F, Cretarola A. Quadratic hedging methods for defaultable claims. Appl Math Optim, 2007, 56(3): 425鈥?43 CrossRef
  5. Biagini F, Cretarola A. Local risk-minimization for defaultable claims with recovery process. Appl Math Optim, 2012, 65(3): 293鈥?14 CrossRef
  6. Bielecki T, Jang I. Portfolio optimization with a defaultable security. Asia-Pacific Finan Markets, 2006, 13: 113鈥?27 CrossRef
  7. Bielecki T, Jeanblanc M, Rutkowski M. Hedging for Defaultable Claims. Lecture Notes in Math, Vol 1847. Berlin: Springer, 2004
  8. Bielecki T, Jeanblanc M, Rutkowski M. Credit Risk Modeling. CSFI Lecture Notes Series. Osaka: Osaka University Press, 2009
  9. Bielecki T, Rutkowski M. Credit Risk: Modeling, Valuation and Hedging. New York: Springer, 2002
  10. Bo L, Wang Y, Yang X. An optimal portfolio problem in a defaultable market. Adv Appl Probab, 2010, 42(3): 689鈥?05 CrossRef
  11. Crandall M G, Ishii H, Lions P L. User鈥檚 guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc, 1992, 27(1): 1鈥?7 CrossRef
  12. Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc, 1983, 277(1): 1鈥?2 CrossRef
  13. Fleming W H, Soner H M. Control Markov Processes and Viscosity Solutions. New York: Springer-Verlag, 1993
  14. Giesecke K. Default and information. J Economic Dyn Control, 2006, 30: 2281鈥?303 CrossRef
  15. Hindy A, Huang C. Optimal consumption and portfolio rules with durability and local substitution. Econometrica, 1993, 61: 85鈥?22 CrossRef
  16. Hou Y, Jin X. Optimal investment with default risk. FAME Research Paper, No 46, Switzerland, 2002
  17. Ishikawa Y. Optimal control problem associated with jump processes. Appl Math Optim, 2004, 50: 21鈥?5 CrossRef
  18. Jang I. Portfolio Optimization with Defaultable Securities. Ph D Thesis. The Univ of Illinois at Chicago, 2005
  19. Jarrow R, Lando D, Yu F. Default risk and diversification: Theory and applications. Math Finan, 2005, 51: 1鈥?6 CrossRef
  20. Jeanblanc M, Rutkowski M. Modelling of default risk: mathematical tools. Department of Mathematics, Universit茅 d鈥橢vry, 2002, Preprint
  21. Jeanblanc M, Yor M, Chesney M. Mathematical Methods for Financial Markets. Berlin: Springer, 2009 CrossRef
  22. Jiao Y, Pham H. Optimal investment with counterparty risk: a default-density modeling approach. Finance Stoch, 2011, 15(4): 725鈥?53 CrossRef
  23. Korn R, Kraft H. Optimal portfolios with defaultable securities: A firm value approach. Int J Theor Appl Finance, 2003, 6: 793鈥?19 CrossRef
  24. Lakner P, Liang W. Optimal investment in a defaultable bond. Math Financ Econ, 2008, 1: 283鈥?10 CrossRef
  25. Lando D. Credit Risk Modeling-Theory and Applications. Princeton and Oxford: Princeton University Press, 2004
  26. Lions P L. Optimal control of diffusion processes and Hamilton-Bellman equations, Part 1: The dynamics programming principle and applications. Comm Partial Differential Equations, 1983, 8(10): 1101鈥?174 CrossRef
  27. Lions P L. Optimal control of diffusion processes and Hamilton-Bellman equations, Part 2: Viscosity solutions and uniqueness. Comm Partial Differential Equations, 1983, 8(11): 1229鈥?276 CrossRef
  28. Protter P. Stochastic Integration and Differential Equations. Berlin: Springer, 1990
  29. Soner H M. Optimal control with state-space constraint, I. SIAM J Control Optim, 1986, 24(3): 552鈥?61 CrossRef
  30. Steffensen K. Asset allocation with contagion and explicit bankruptcy procedures. J Math Econom, 2009, 45: 147鈥?67 CrossRef
  
• 作者单位：Lijun Bo (1)
  Yongjin Wang (2)
  Xuewei Yang (3)
  
  1. Department of Mathematics, Xidian University, Xi鈥檃n, 710071, China
  2. School of Business, Nankai University, Tianjin, 300071, China
  3. School of Management and Engineering, Nanjing University, Nanjing, 210093, China
  
• ISSN：1673-3576

文摘

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi-Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.