Optimal Trend Estimation in Geometric Asset Price Models

In the general geometric asset price model, the asset price P (t) at time t satisfies the relation P (t) = P0 · eα·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.