Optimal trend estimation in geometric asset price models

Michael Weba

Discussiones Mathematicae Probability and Statistics (2005)

  • Volume: 25, Issue: 1, page 51-70
  • ISSN: 1509-9423

Abstract

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In the general geometric asset price model, the asset price P(t) at time t satisfies the relation P ( t ) = P ₀ · e α · f ( t ) + σ · F ( t ) , t ∈ [0,T], where f is a deterministic trend function, the stochastic process F describes the random fluctuations of the market, α is the trend coefficient, and σ denotes the volatility. The paper examines the problem of optimal trend estimation by utilizing the concept of kernel reproducing Hilbert spaces. It characterizes the class of trend functions with the property that the trend coefficient can be estimated consistently. Furthermore, explicit formulae for the best linear unbiased estimator α̂ of α and representations for the variance of α̂ are derived. The results do not require assumptions on finite-dimensional distributions and allow of jump processes as well as exogeneous shocks. .

How to cite

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Michael Weba. "Optimal trend estimation in geometric asset price models." Discussiones Mathematicae Probability and Statistics 25.1 (2005): 51-70. <http://eudml.org/doc/287694>.

@article{MichaelWeba2005,
abstract = {In the general geometric asset price model, the asset price P(t) at time t satisfies the relation $P(t) = P₀ · e^\{α·f(t) + σ·F(t)\}$, t ∈ [0,T], where f is a deterministic trend function, the stochastic process F describes the random fluctuations of the market, α is the trend coefficient, and σ denotes the volatility. The paper examines the problem of optimal trend estimation by utilizing the concept of kernel reproducing Hilbert spaces. It characterizes the class of trend functions with the property that the trend coefficient can be estimated consistently. Furthermore, explicit formulae for the best linear unbiased estimator α̂ of α and representations for the variance of α̂ are derived. The results do not require assumptions on finite-dimensional distributions and allow of jump processes as well as exogeneous shocks. .},
author = {Michael Weba},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {geometric asset price model; trend estimation; Wiener process; Ornstein-Uhlenbeck process; kernel reproducing Hilbert space; exogeneous shocks; compound Poisson process; reproducing kernel Hilbert space},
language = {eng},
number = {1},
pages = {51-70},
title = {Optimal trend estimation in geometric asset price models},
url = {http://eudml.org/doc/287694},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Michael Weba
TI - Optimal trend estimation in geometric asset price models
JO - Discussiones Mathematicae Probability and Statistics
PY - 2005
VL - 25
IS - 1
SP - 51
EP - 70
AB - In the general geometric asset price model, the asset price P(t) at time t satisfies the relation $P(t) = P₀ · e^{α·f(t) + σ·F(t)}$, t ∈ [0,T], where f is a deterministic trend function, the stochastic process F describes the random fluctuations of the market, α is the trend coefficient, and σ denotes the volatility. The paper examines the problem of optimal trend estimation by utilizing the concept of kernel reproducing Hilbert spaces. It characterizes the class of trend functions with the property that the trend coefficient can be estimated consistently. Furthermore, explicit formulae for the best linear unbiased estimator α̂ of α and representations for the variance of α̂ are derived. The results do not require assumptions on finite-dimensional distributions and allow of jump processes as well as exogeneous shocks. .
LA - eng
KW - geometric asset price model; trend estimation; Wiener process; Ornstein-Uhlenbeck process; kernel reproducing Hilbert space; exogeneous shocks; compound Poisson process; reproducing kernel Hilbert space
UR - http://eudml.org/doc/287694
ER -

References

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  1. [1] M. Atteia, Hilbertian Kernels and Spline Functions, North-Holland, Amsterdam 1992. 
  2. [2] S. Cambanis, Sampling Designs for Time Series, Time Series in the Time Domain, Handbook of Statistics 5 (1985), (Eds. E.J. Hannan, P.R. Krishnaiah, and M.M. Rao), North-Holland, Amsterdam, 337-362. 
  3. [3] J.Y. Campbell, A.W. Lo and A.C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, Princeton 1997. Zbl0927.62113
  4. [4] E. Eberlein and U. Keller, Hyperbolic Distributions in Finance, Bernoulli 1 (1995), 281-299. Zbl0836.62107
  5. [5] I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, Springer, New York 1998. Zbl0941.91032
  6. [6] M. Loève, Probability Theory II, (4th Edition), Springer, New York 1978. 
  7. [7] R. Merton, On Estimating the Expected Return on the Market: An Exploratory Investigation, Journal of Financial Economics 8 (1980), 323-361. 
  8. [8] E. Parzen, Regression Analysis of Continuous Parameter Time Series, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1 (1961), (Ed. J. Neyman), University of California Press, 469-489. Zbl0107.13802
  9. [9] J. Sacks and D. Ylvisaker, Designs for Regression Problems with Correlated Errors, The Annals of Mathematical Statistics 37 (1966), 66-89. 
  10. [10] J. Sacks and D. Ylvisaker, Designs for Regression Problems with Correlated Errors: Many Parameters, The Annals of Mathematical Statistics 39 (1968), 46-69. Zbl0165.21505
  11. [11] J. Sacks and D. Ylvisaker, Designs for Regression Problems with Correlated Errors III, The Annals of Mathematical Statistics 41 (1970), 2057-2074. Zbl0234.62025
  12. [12] R.J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley 2002. Zbl1001.62005
  13. [13] Y. Su and S. Cambanis, Sampling Designs for Estimation of a Random Process, Stochastic Processes and Their Applications 46 (1993), 47-89. Zbl0771.62065

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