Testing Problems for the Density Process for fBm with Different Drifts
As we have seen in Subsection 5.2.2, the form of geometric fBm (5.2.6) depends on the kind of integral that is used in its calculations: if we use the Riemann–Stieltjes integral,
then \(S_t^{\left( 1 \right)} = S_0^{\left( 1 \right)} \exp \left\{ {\mu t + \sigma B_t^H } \right\},\) and if the behavior of geometric process is guided by the Wick integral,
then \(S_t^{\left( 2 \right)} = S_0^{\left( 2 \right)} \exp \left\{ {\mu t + \sigma B_t^H - \frac{1}{2}\sigma ^2 t^{2H} } \right\}\) So, the natural question arises: what trend actually has geometric fBm? This question was considered in the paper (KMV05), and here we present a solution of this problem. In what follows the notation \(X_n = o_P \left( 1 \right)\) means that \(X_n \rightarrow{P}0,X_n = O_P \left( 1 \right)\) means that \(\mathop {\lim }\limits_{C \to \infty } \mathop {\lim \sup }\limits_n P\left\{ {\left| {X_n } \right| \ge C} \right\} = 0.\) Assume that \(H \in \left( {1/2,1} \right)\). For a fixed \(\mu \in {\rm R}\) let \(P_{\mu ,\sigma } \) be the distribution of the process
in the space \(C_{\left[ {0,T} \right]} \) of continuous functions. Similarly, \(P_{\mu ,\sigma } \) is the distribution of the process
in the space \(C_{\left[ {0,T} \right]} \)
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(2008). Statistical Inference with Fractional Brownian Motion. In: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol 1929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75873-0_6
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