By G. Kallianpur, D. Kölzow

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**Extra resources for Measure Theory Applications to Stochastic Analysis**

**Sample text**

2), since Hu ⊆ 32 3. 2) is Hu -measurable. By the re-parametrization α = − 12 θ2 , we can write √ dP¯θu = exp(ατu + −2αt), α ≤ 0. 3) u ¯ dP Under P , the stochastic process √ {τu } has independent increments, and τ1 has cumulant transform − −2s with domain s ∈ (−∞, 0]. Hence the restriction of the elements in {Pθ : θ ≥ 0} to σ(Hu : u ≥ 0) equals the exponential family of processes with independent increments obtained from {τu } and P as discussed in Chapter 2. Another example illustrating the signiﬁcance of the ﬁltration is the following.

M, are real-valued right-continuous stochastic processes with limits from the left and adapted to {Ft }. 6 29 Diﬀusion processes with jumps An important generalization of the diﬀusion processes is the class of diﬀusion processes with jumps. 1) t > 0, X0θ = x0 . Here W is a d-dimensional standard Wiener process, and θT = (θ1T , θ2 , θ3T ), θ1 ∈ Θ1 ⊆ IRk1 , θ2 ∈ IR, θ3 ∈ Θ3 ⊆ IRk3 . The dimensions of the process X θ and of the functions at , bt , and ct are as in the previous section (with k replaced by k1 ), and dt is an invertible d × d-matrix.

Then a process of the form Xt = exp(Wt + θt), θ ∈ IR, is called a geometric Brownian motion. The likelihood function corresponding to observation of Xt in [0, t] is Lt (θ) = exp[(θ + 12 ) log(Xt ) + 12 (θ + 12 )2 t], θ ∈ IR. 2 Exponential families of stochastic processes with a non-empty kernel 41 We see that int Γ = IR = ∅, and indeed, Bt = log(Xt ) = Wt + θt has independent increments for all θ ∈ IR. 6) with X0 = 1, as follows by a straightforward application of Ito’s formula. ✷ We see that Xt does not have independent increments.