Skip to main content
SpringerLink
Log in

Non-ideal Brownian motion, generalized Langevin Equation and its application to the security market

  • Published:
Financial Engineering and the Japanese Markets Aims and scope Submit manuscript

Abstract

Brownian motion has been extensively applied in the field of mathematical finance in modeling the stochastic processes of returns on securities. In this paper basic and generalized Langevin Equations with memory are used to augment Brownian motion to capture the well stylized facts of the financial market that frictions and imperfect information exist. The operator method of Fourier-Laplace transform with an appropriate kernel (influence function) is used to circumvent the difficulty associated with solving a time dependent nonlinear differential Equation, and a practical computational method is proposed.

From the Langevin Equation, autocorrelation of the return process and the deviation of the return distribution from an ideal Brownian motion are extracted. It is also proven that the time-dependent differential Equation has a unique solution and that it is much more generalized than a martingale Brownian motion functional.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alder, B.J. and Wainwright, T.E. (1967), ‘Velocity Autocorrelations for Hard Spheres’,Physical Review Letters,18, 989–999.

    Google Scholar 

  2. Bollerslev, T., Chow, R.Y., Jarayaman, N. and Kroner, K.F. (1990), ‘ARCH Modelling in Finance: A Selective Review of the Theory and Empirical Evidence’, Working Paper.

  3. Ahlfors, L.V. (1966), Complex Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, Kogakusya.

    Google Scholar 

  4. Case, M.A. (1971), ‘Velocity Fluctuation of a Body in a Fluid’,The Physics of Fluids,14(10), 2091–2095.

    Google Scholar 

  5. Cole, C.S. and Reichenstein, W. (1994), ‘Forecasting Interest Rates with Eurodollar Futures Rates’,Journal of Futures Market,10(1), 37–50.

    Google Scholar 

  6. Engle, R.F. (1982), ‘Autoregressive Conditional Heteroskedastisity with Estimates of the Variance of United Kingdom Inflation’,Econometrica,50, 987–1007.

    Google Scholar 

  7. Fama, E. (1965), ‘The Behavior of Stock Market Prices’,Journal of Business,38, 34–105.

    Google Scholar 

  8. Feder, J. (1988), Fractals, Plenum Press.

  9. Granger, C.W.J. and Newbold, P (1977), Forecasting Economic Time Series, Academic Press.

  10. Heston, S.L. (1993), ‘A Closed Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options’,Review of Financial Studies,6(2), 327–343.

    Google Scholar 

  11. Hori, Jyunichi (1977), The Langevin Equation (in Japanese), Iwanami-Syoten.

  12. Hsieh, D.A. (1989), ‘Testing for Nonlinear Dependence in Daily Foreign Exchange Rate’,Journal of Business,62(3), 339–368.

    Google Scholar 

  13. Kannan, D. and Bharucha-Reid, A.T. (1972), ‘Random Integral Equation Formulation of a Generalized Langevin Equation’,Journal of Statistical Physics,5(3), 209–233.

    Google Scholar 

  14. Kariya, T. (1993), Quantitative Methods For Portfolio Analysis, Theory And Decision Library (Series B), Kluwer Academic Publishers.

  15. Kariya, T. (1990), Basic Quantitative Course in Portfolio Analysis (in Japanese), Toyokeizai-Shinpou-Sya.

  16. Kariya, T., Tsukuda, Y., Maru, J., Matsue, Y. and Omaki, K. (1995), ‘An Extensive Analysis on the Japanese Markets via S. Taylor's Model’,Financial Engineering and the Japanese Markets,2(1), 15–86.

    Google Scholar 

  17. Karandikar, R.L. and Rachev, S.T. (1993), ‘A Generalized Binomial Model and Option Formulae for Subordinated Stock-Price Processes’, Technical Report No. 242, Department of Statistics and Applied Probability, Univ. of California, Santa Barbara, February.

    Google Scholar 

  18. Kato, K. (1990), Stock Price Changes and Anomaly (in Japanese), Nihon-Keizaishinbun-Sya.

  19. Kitahara, K. (1994), Science in Non-Equilibrium Systems II: Statistical Physics in Relaxation Process (in Japanese), Science Series, Kodan-Sya.

  20. Kondo, J. (1961), Integral Equation, Baifukan.

  21. Konno, S., Shirakawa, S. and Yamazaki, H (1993), ‘A Mean-Absolute Deviation-Skewness Portfolio Optimization Model’,Annals of Operations Research,45, 205–220.

    Google Scholar 

  22. Kubo, R. (1979), ‘Irreversible Process and Stochastic Process: Statistical Dynamics on Stochastic Process and Open System (in Japanese)’,Memorandum of Research Institute of Mathematical Science,367, 50–93.

    Google Scholar 

  23. Kubo, R., Toda, M. and Hashimoto, N. (1985), Statistical Physics II, Springer-Verlag.

  24. Kunitomo, N. (1992), ‘Long Memory and Geometric Brownian Motion in Security Market Models’,Discussion Paper 92-F-12 (University of Tokyo).

  25. Landau, L.D. and Lifshitz, E.M. (1959), Fluid Mechanics, Pergamon.

  26. Loretan, M. and Phillips, P.C.B. (1994), ‘Testing the Covariance Stationarity of Heavy-tailed Time Series’,Journal of Empirical Finance,1(2), 211–248.

    Google Scholar 

  27. Mandelbrot, B.B. (1961), ‘Stable Paretian Random functions and the Multiplicative Variation of Income’,Econometrica,29, 517–543.

    Google Scholar 

  28. Mandelbrot, B.B. (1963), ‘The Variation of Some Speculative Prices’,The Journal of Business,26, 394–419.

    Google Scholar 

  29. Mandelbrot, B.B. and Van Ness, J.W. (1968), ‘Fractional Brownian Motions, Fractional Noises and Applications’,SIAM Review,10, 422–437.

    Google Scholar 

  30. Mandelbrot, B.B. (1971), ‘When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models’,The Review of Economics and Statistics, August,53, 225–236.

    Google Scholar 

  31. Miura, R. (1989), ‘Stock Price Changes by Mixed Normal Model, and Option Pricing (in Japanese)’,Hitotsubashi Ronsou,102(5), 620–644.

    Google Scholar 

  32. Obayashi, K. (1985), ‘Non-Markovian Property and Nonlinear Relaxation (in Japanese)’,Physics Monthly,6(8), 440–444.

    Google Scholar 

  33. Ogura, N. (1985), Theory of Stochastic Process for Physics and Engineering (in Japanese), Corona Publishing.

  34. Okabe, Y. (1992), ‘Characterization of Stochastic Process (in Japanese)’,Mathematical Science,5, 11–17.

    Google Scholar 

  35. Okabe, Y. (1986), ‘On KMO-Langevin Equations for Stationary Gaussian Processes with T-Positive’,Journal of the Faculty of Science, University of Tokyo, Sec. 1A,33(1), 1–56.

    Google Scholar 

  36. Okabe, Y. and Inoue, A. (1992), ‘On the Exponential Decay of the Correlation Functions for KMO-Langevin Equation’,Journal of Mathematics,18(1), 13–24.

    Google Scholar 

  37. Okabe, Y. (1986), ‘On the Theory of Brownian Motion with the Alder-Wainwright Effect’,Journal of Statistical Physics,45(5/6), 953–981.

    Google Scholar 

  38. Peters, E.E. (1989) ‘Fractal Structure in the Capital Markets (in Japanese)’,Security Analysts Journal,27(27), 58–66.

    Google Scholar 

  39. Spiegel, M.R. (1965), Theory and Problems of Laplace Transforms, Schaum's Outline Series, McGraw-Hill.

  40. Sunahara, Y (1982), Theory of Stochastic Systems 11 (in Japanese), Asakura-Syoten.

  41. Suzuki, M. (1981), ‘Passage from an Initial Unstable State to Final Stable State’,Advances in Chemical Physics,46, 195–279.

    Google Scholar 

  42. Suzuki, M. (1992), ‘Fokker-Plank Equation (in Japanese)’,Mathematical Science (extra issue), 128–137.

  43. Takahashi, M. (1992), Valuation of Derivative Assets Utilizing Hurst Exponent (in Japanese), Master Thesis in University of Tsukuba.

  44. Taylor, S.J. (1986), Modelling Financial Time Series, John Wiley and Sons Ltd.

  45. Terasawa, K. (1974), Introduction to Mathematics for Natural Scientists (in Japanese), Iwanami-Syoten.

  46. Toda, M. and Kubo, R. (1972), ‘Statistical Physics (in Japanese),’ Basic Theories in Modern Physics,6, Iwanami-Syoten.

  47. Yajima, Y. (1989), ‘A Central Limit Theorem of Fourier Transforms of Strongly Dependent Stationary Processes’,Journal of Time Series Analysis,10(4), 375–383.

    Google Scholar 

  48. Yamada, N. and Kunieda, T. (1971), Laplace Transformation and Operational Calculus (in Japanese), Corona Publishing.

Download references

Author information

Authors and Affiliations

  1. Institute of Socio-Economic Planning, University of Tsukuba, Tsukuba-shi, Japan

    Masafumi Takahashi

Authors
  1. Masafumi Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takahashi, M. Non-ideal Brownian motion, generalized Langevin Equation and its application to the security market. Financial Engineering and the Japanese Markets 3, 87–119 (1996). https://doi.org/10.1007/BF00868082

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00868082

Key words

Search

Navigation