Exponential Families of Stochastic Processes by Uwe Küchler

By Uwe Küchler

This ebook presents a entire account of the statistical concept of exponential households of stochastic methods. The booklet reports the development within the box revamped the final ten years or so by way of the authors, of the top specialists within the box, and a number of other researchers. the speculation is utilized to a huge spectrum of examples. The statistical ideas are defined rigorously in order that probabilists with just a simple heritage in facts can use the booklet to get into statistical inference for stochastic strategies. workouts are incorporated to make the ebook worthy for a sophisticated graduate direction.

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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.

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