Statistical Inference with Fractional Brownian Motion

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

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,
$$S_t^{\left( 1 \right)} = S_0^{\left( 1 \right)} + \mu \int_0^1 {S_s^{\left( 1 \right)} ds} + \sigma \int_0^t {S_s^{\left( 1 \right)} dB_s^H ,} $$
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,
$$S_t^{\left( 2 \right)} = S_0^{\left( 2 \right)} + \mu \int_0^2 {S_s^{\left( 2 \right)} ds} + \sigma \int_0^t {S_s^{\left( 2 \right)} \diamondsuit dB_s^H ,} $$
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
$$X_t : = \sigma B_t^H + \mu t - \frac{{\sigma ^2 }}{2}t^{2H} ,0 \le t \le T$$
in the space \(C_{\left[ {0,T} \right]} \) of continuous functions. Similarly, \(P_{\mu ,\sigma } \) is the distribution of the process
$$X_t : = \sigma B_t^H + \mu t,0 \le t \le T$$
in the space \(C_{\left[ {0,T} \right]} \)


Likelihood Ratio Probability Measure Maximum Likelihood Estimate Statistical Inference Wiener Process 
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