Frontiers of Mathematics in China >
Optimal portfolio and consumption selection with default risk
Received date: 25 Oct 2010
Accepted date: 21 May 2012
Published date: 01 Dec 2012
Copyright
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.
Lijun BO , Yongjin WANG , Xuewei YANG . Optimal portfolio and consumption selection with default risk[J]. Frontiers of Mathematics in China, 2012 , 7(6) : 1019 -1042 . DOI: 10.1007/s11464-012-0224-3
1 |
Atar R, Budhiraja A, Williams R J. HJB equations for certain singularly controlled diffusions. Ann Appl Probab, 2007, 17(5-6): 1745-1776
|
2 |
Bélanger A, Shreve S, Wong D. A general framework for pricing credit risk. Math Finance, 2004, 14(3): 317-350
|
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: 275-303
|
4 |
Biagini F, Cretarola A. Quadratic hedging methods for defaultable claims. Appl Math Optim, 2007, 56(3): 425-443
|
5 |
Biagini F, Cretarola A. Local risk-minimization for defaultable claims with recovery process. Appl Math Optim, 2012, 65(3): 293-314
|
6 |
Bielecki T, Jang I. Portfolio optimization with a defaultable security. Asia-Pacific Finan Markets, 2006, 13: 113-127
|
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): 689-705
|
11 |
Crandall M G, Ishii H, Lions P L. User's guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc, 1992, 27(1): 1-67
|
12 |
Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc, 1983, 277(1): 1-42
|
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: 2281-2303
|
15 |
Hindy A, Huang C. Optimal consumption and portfolio rules with durability and local substitution. Econometrica, 1993, 61: 85-122
|
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: 21-65
|
18 |
Jang I. Portfolio Optimization with Defaultable Securities. <DissertationTip/>. 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: 1-26
|
20 |
Jeanblanc M, Rutkowski M. Modelling of default risk: mathematical tools. Department of Mathematics, Université d'Evry, 2002, Preprint
|
21 |
Jeanblanc M, Yor M, Chesney M. Mathematical Methods for Financial Markets. Berlin: Springer, 2009
|
22 |
Jiao Y, Pham H. Optimal investment with counterparty risk: a default-density modeling approach. Finance Stoch, 2011, 15(4): 725-753
|
23 |
Korn R, Kraft H. Optimal portfolios with defaultable securities: A firm value approach. Int J Theor Appl Finance, 2003, 6: 793-819
|
24 |
Lakner P, Liang W. Optimal investment in a defaultable bond. Math Financ Econ, 2008, 1: 283-310
|
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): 1101-1174
|
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): 1229-1276
|
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): 552-561
|
30 |
Steffensen K. Asset allocation with contagion and explicit bankruptcy procedures. J Math Econom, 2009, 45: 147-167
|
/
〈 | 〉 |