Application of E^p-Stability to Impulsive Financial Model

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Benjamin Oyediran Oyelami, Sam Olatunji Ale

Abstract

In this paper, we consider an impulsive stochastic model for an investment with production and saving profiles. The conditions for financial growth for the investment are investigated under impulsive action and results are obtained using the quantitative and Ep stability methods. The impulsive stochastic differential equation considered is assumed to be driven by a process with jump and non-linear gestation properties. One of the results established shows that, in the long run, it is impossible for a financial investment to grow or dominates the prescribed average financial investment but has a threshold value for which the investment cannot grow beyond. It is also established that an $E^{p}-$ stable investment vector can be found which allows financial growth but this vector must be constrained to be in a given invariant set:It is advisable for the saving and depreciation to satisfy certain growth rates for proper income and investment growths.

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References

  1. Ale, S.O. and Oyelami, B.O., Impulsive systems and Applications. Int. J. Math. Edu. Sci. Technol., 2000, Vol. No. 4, 539-544.
  2. Black, F. and Scholes, M. , The pricing of Options and Corporate Liabilities. Journal of Political Economics, 81, 637-654(1973).
  3. Gardiner, C.W. and Zoller, P., Quantum noise: A handbook of Markovian and NonMarkovian Quantum Stochastic Methods with Applications to Quantum Optics. Second edition. Berlin, Springer-Verlag 2000.
  4. Guilia Lori, Financial Derivatives. ICTP Lecture notes on Financial Mathematics, 2007.
  5. Jacob, J. and Shiriaev, A.N., Limit Theorems for Stochastic Processes. Springer-Verlag, 2nd edition, 2003.
  6. Jacqueline Terra Moura Marins and Eduardo Saliby, Crdit Risk Monte Carlos Simulation using Creditmetrics Model: The joint use of importance sampling and descriptive sample. Banco Central Do Brasil, March 2007. Working paper series 132.
  7. Joao, P. Hespanha and Andrew R. Teel., Stochastic Impulsive Systems Driven by Renewal Processes Extended version. Proc. Int. Sympo. in mathematical theory,2006, 1-26., Available on , http://www.ece.ucsb.edu/hespanhapublished/mtns06sto-ncs.pds.
  8. Juan J. Neito, , Periodic-Boundary Value Problems for first order Impulsive Ordinary Differential Equations, Nonlinear Anal. 51(2002), 1223-1232.
  9. Hull, J.C. and While A., , Efficient procedures for valuing European and American pathdependent options. Journal of Derivatives, 1, 21-31, 1993.
  10. Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S., Theorem of Impulsive Differential Equations (Singapore; World Scientific).
  11. Lakshmikantham, V., Stability of Moving Invariant Sets and Uncertain Dynamical System, Discrete Contin, Dynamic System, added vol. II(1998), 24-31.
  12. Laksmikantham, V. and Z. Drici,, Stability of conditionally Invariant Sets and Controlled Uncertain Dynamical Systems with Time Scales. Math. Prob. Eng. 11 (1995), 1-10.
  13. Marco Airoldi and Mediobanca., A Perturbative Moment Approach to Option Pricing., ArXiv:cond-mat/0401503 v1, 2004. Preprint.
  14. Oyelami, B.O., Ale, S.O. and Sesay, M.S., On Existence of Solutions and Stability with respect to Invariant Sets for Impulsive Differential Equations with variable times. Advances in Differential Equations and Control Processes, 2008.
  15. Oyelami, B.O., On Military Model for Impulsive Reinforcement functions using Exclusion and Marginalization techniques. Nonlinear Analysis 35(1999), 947-958.
  16. Oyelami, B.O., Impulsive Systems and Applications to some Models. Ph.D. Thesis, Abubakar Tafawa Balewa University of Technology, Bauchi, Nigeria. 1999.
  17. Simeonov, P.S. and Bainov, D.D., Systems with Impulsive Effects; Stability, Theory and Applications (Ellis Horwood), 1989.
  18. Sullivan, M.A., Pricing discretely monitored Barrier Options. Journal of Computational Finance, Vol. 3, No. 4, Summer, 35-52(2000).
  19. Umut Cetin and Rogers, L.C.G., Modeling Liquidity Effect in Discrete Time. Mathematical Finance, Vol. 17, No. 1 (January, 2007), 15-29.