By Finkenstadt B. F.

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73) in Chapter 1. s. [Pθ0 ] i+1 i where (n) =T 0 = t0(n) < t1(n) < · · · < tn−1 is a partition of the interval [0, T ] such that (n) − ti(n) | → 0 sup |ti+1 0≤i≤n−1 as n → ∞. ) is an unknown constant σ , the above property can be used to obtain a strongly consistent estimator of σ 2 based on the continuous 50 STATISTICAL INFERENCE FOR FRACTIONAL DIFFUSION PROCESSES observation of the process X over the interval [0, T ]. Hereafter we assume that the non random function σ (t) is known. We now discuss the problem of estimation of the parameter θ on the basis of the observation of the process X or equivalently of the process Z in the interval [0, T ].

101) FRACTIONAL BROWNIAN MOTION 31 by an application of Ito’s lemma (cf. Prakasa Rao (1999a)). Such a process is called the Ornstein–Uhlenbeck process. 102) where λ and σ 2 are constants. The existence and uniqueness of the solution of this SDE are discussed in Cheridito et al . (2003). This process is called a fractional Ornstein–Uhlenbeck-type process. 24: Let {WtH , −∞ < t < ∞} be fBm with index H ∈ (0, 1] deﬁned on a probability space ( , F, P ) and X(0, ω) = η(ω) ∈ R. Let −∞ ≤ a < ∞ and λ, σ > 0.

83) and C1 = C0 B 3 3 − H, − H . 85) and w(t, u) = C0 u 2 −H (t − u) 2 −H . 86) Norros et al . 87) is well deﬁned as a pathwise integral and is a martingale with respect to the ﬁltration {Ft , t ≥ 0} generated by the process W H . The quadratic variation of t 2−2H . 88) 0 is a Wiener process W adapted to the same ﬁltration. 89) (ii) and there exists M > 0 such that |S(t, x)| ≤ M(1 + |x|), x ∈ R, 0 ≤ t ≤ T . 92) has a unique solution {Xt , 0 ≤ t ≤ T } and the sample paths of this process belong to C H − [0, T ] with probability one.

### A stochastic model for extinction and recurrence of epidemics estimation and inference for measles o by Finkenstadt B. F.

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