Discussiones Mathematicae Probability and Statistics 25(1) (2005) 51-70

OPTIMAL TREND ESTIMATION IN GEOMETRIC ASSET PRICE MODELS

Michael Weba

University of Applied Sciences, Fulda
Marquardstr. 35, D-36039 Fulda, Germany

Abstract

In the general geometric asset price model, the asset price P(t) at time t satisfies the relation
P(t) = P0 ·e  a·f(t) + s·F(t)  ,     t Î [0,T],
where f is a deterministic trend function, the stochastic process F describes the random fluctuations of the market, a is the trend coefficient, and s 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 [^(a)] of a and representations for the variance of [^(a)] are derived.

The results do not require assumptions on finite-dimensional distributions and allow of jump processes as well as exogeneous shocks.

Keywords: geometric asset price model, trend estimation, Wiener process, Ornstein-Uhlenbeck process, kernel reproducing Hilbert space, exogeneous shocks, compound Poisson process.

2000 Mathematics Subject Classification: Primary 62J05, 62K05, 62M10; Secondary 62P05, 62P20.

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Received 15 February 2004