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1993)). Note that if f or g is a smooth function on a ﬁnite interval (a, b), the Lebesgue–Stieltjes integral can be written in the form b b f (x) dg(x) = a f (x)g (x)dx a or b a b f (x) dg(x) = − f (x)g(x)dx + f (b−)g(b−) − f (a+)g(a+). a Here f (a+) = limδ 0 f (a + δ) and g(b−) = limδ 0 f (b−δ) whenever the limits exist. The main idea of Zahle’s approach is to replace the ordinary derivatives by the fractional derivatives. Let fa+ (x) = (f (x) − f (a+))1(a,b) (x) and gb− (x) = (g(x) − g(b−))1(a,b) (x) where 1(a,b) (x) = 1 if x ∈ (a, b) and 1(a,b) (x) = 0 otherwise.

The problem of interest is the estimation of the parameter θ Statistical Inference for Fractional Diffusion Processes B. L. S. Prakasa Rao 2010 John Wiley & Sons, Ltd 46 STATISTICAL INFERENCE FOR FRACTIONAL DIFFUSION PROCESSES based on the observation or the data {Xt , 0 ≤ t ≤ T }. The problem of estimation of the Hurst index H is also very important. However, we will assume that H is known in the following discussion. We will discuss the problem of estimation of the Hurst index brieﬂy in Chapter 9.

Let W be standard Brownian motion. 42) where fH, 1 (t, u) = 2 C1 (H ) 1 −H (H + 12 ) ( 32 − H ) ((t − u)+2 1 −H − (−u)+2 ). 43) Let FH,t denote the σ -algebra generated by the process {W H (s), 0 ≤ s ≤ t} and F 1 ,t denote the σ -algebra generated by the process {W (s), 0 ≤ s ≤ t}. 44) holds for each t ∈ R almost everywhere and hence the σ -algebras FH,t and F 1 ,t 2 are the same up to sets of measure zero for t > 0. 44) does not hold for t < 0. s. but with zero quadratic variation whenever H > 12 .