RESEARCH ARTICLE

Optimal control of a big financial company with debt liability under bankrupt probability constraints

  • Zongxia LIANG ,
  • Bin SUN
Expand
  • Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 06 Aug 2010

Accepted date: 13 Aug 2010

Published date: 01 Dec 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company’s bankruptcy soon. In order to protect the shareholders’ profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, partial differential equation and variational inequality approach. Moreover, we establish a riskbased capital standard to ensure the capital requirement can cover the total given risk by numerical analysis, and give reasonable economic interpretation for the results.

Cite this article

Zongxia LIANG , Bin SUN . Optimal control of a big financial company with debt liability under bankrupt probability constraints[J]. Frontiers of Mathematics in China, 2011 , 6(6) : 1095 -1130 . DOI: 10.1007/s11464-011-0120-2

1
Asmussen S, Høgaard B, Taksar M. Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation. Finance and Stochastics, 2000, 4: 199-324

DOI

2
Asmussen S, Taksar M. Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 1997, 20: 1-15

DOI

3
Avanzi B. Strategies for dividend distribution: A review. North American Actuarial Journal, 2009, 13: 217-251

4
Borodin A N, Salminen P. Handbook of Brownian Motion: Facts and Formulae. Boston: irkhäuser, 2002

DOI

5
Choulli T, Taksar M, Zhou Xunyu. Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction. Quantitative Finance, 2001, 1: 573-596

DOI

6
Choulli T, Taksar M, Zhou Xunyu. An optimal diffusion model of a company with constraints on risk control. SIAM Journal on Control and Optimization, 2003, 41: 1946-1979

DOI

7
Choulli T, Taksar M, Zhou Xunyu. Interplay between dividend rate and business constraints for a financial corporation. The Annals of Applied Probability, 2004, 14: 1810-1837

DOI

8
Emanuel D, Harrison J, Taylor A. A diffusion approximation for the ruin probability with compounding assets. Scandinavian Actuarial Journal, 1975, 75: 240-247

9
Grandell J. A class of approximations of ruin probabilities. Scandinavian Actuarial Journal, 1977, 77(Suppl): 37-52

10
Grandell J. A remark on a class of approximations of ruin probabilities. Scandinavian Actuarial Journal, 1978, 78: 77-78

11
Grandell J. Aspect of Risk Theory. New York: Springer, 1990

12
Guo X, Liu J, Zhou X. A constrained nonlinear regular-singular stochastic control problem, with application. Stochastic Processes and Their Applications, 2004, 109: 167-187

DOI

13
Harrison J. Brownian Motion and Stochastic Flow Systems. New York: Wiley, 1986

14
He Lin, Hou Ping, Liang Zongxia. Optimal financing and dividend control of the insurance company with proportional reinsurance policy under solvency constraints. Insurance: Mathematics and Economics, 2008, 43: 474-479

DOI

15
He Lin, Liang Zongxia. Optimal dividend control of the insurance company with proportional reinsurance policy under solvency constraints. Insurance: Mathematics and Economics, 2008, 42: 976-983

DOI

16
He Lin, Liang Zongxia. Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance: Mathematics and Economics, 2009, 44: 88-94

DOI

17
Høgaard B, Taksar M. Controlling risk exposure and dividends payout schemes: Insurance company example. Mathematical Finance, 1999, 9: 153-182

DOI

18
Høgaard B, Taksar M. Optimal risk control for a large corporation in the presence of returns on investments. Finance and Stochastics, 2001, 5: 527-547

DOI

19
Iglehart D. Diffusion approximations in collective risk theory. Journal of Applied Probability, 1969, 6: 285-292

DOI

20
Ikeda N, Watanabe S. A comparison theorem for solutions of stochastic differential equations and its applications. Osaka Journal of Mathematics, 1977, 14: 619-633

21
Liang Zongxia, Huang Jianping. Optimal dividend and investing control of a insurance company with higher solvency constraints. arXiv:1005.1360

22
Liang Zongxia, Yao Jicheng. Nonlinear optimal stochastic control of large insurance company with insolvency probability constraints. arXiv:1005.1361

23
Liang Zongxia, Yao Jicheng. Optimal dividend policy of a large insurance company with positive transaction cost under higher solvency and security. arXiv:1005.1356

24
Lions P L, Sznitman A S. Stochastic differential equations with reflecting boundary conditions. Communications on Pure and Applied Mathematics1984, 37: 511-537

DOI

25
Paulsen J. Optimal dividend payouts for solvency constraints. Finance and Stochastics, 2003, 7: 457-473

DOI

26
Schmidli H. Diffusion approximations for a risk process with the possibility of borrowing and interest. Communications in Statistics. Stochastic Models, 1994, 10: 365-388

DOI

27
Taksar M. Optimal risk and dividend distribution control models for an insurance company. Mathematical Methods of Operations Research, 2000, 1: 1-42

DOI

28
Taksar M, Zhou Xunyu. Optimal risk and dividend control for a company with a debt liability. Insurance: Mathematics and Economics, 2008, 22: 105-122

DOI

Options
Outlines

/