Download PDF by Jie Xiong: An Introduction to Stochastic Filtering Theory

By Jie Xiong

ISBN-10: 0199219702

ISBN-13: 9780199219704

Stochastic Filtering Theory makes use of likelihood instruments to estimate unobservable stochastic procedures that come up in lots of utilized fields together with communique, target-tracking, and mathematical finance. As a subject matter, Stochastic Filtering concept has stepped forward quickly lately. for instance, the (branching) particle procedure illustration of the optimum clear out has been largely studied to hunt better numerical approximations of the optimum filter out; the soundness of the filter out with "incorrect" preliminary nation, in addition to the long term habit of the optimum filter out, has attracted the eye of many researchers; and even supposing nonetheless in its infancy, the learn of singular filtering versions has yielded fascinating effects. during this textual content, Jie Xiong introduces the reader to the fundamentals of Stochastic Filtering concept earlier than protecting those key fresh advances. The textual content is written in a method appropriate for graduates in arithmetic and engineering with a historical past in uncomplicated chance.

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Extra resources for An Introduction to Stochastic Filtering Theory

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S. and in L1 ( ). 6) where F∞ = σ (∪n Fn ) ≡ ∨n Fn . Proof By Jensen’s inequality, we have |Xn | ≤ E |Y| Fn , and hence E |Xn |1|Xn |>λ ≤ E E |Y| Fn 1|Xn |>λ = E |Y|1|Xn |>λ ≤ E |Y|1|Y|>λ + λ P (|Xn | > λ) ≤ E |Y|1|Y|>λ + λ−1 λ E (|Xn |) ≤ E |Y|1|Y|>λ + λ−1 λ E (|Y|) , where λ is an arbitrary constant. Then lim sup sup E |Xn |1|Xn |>λ ≤ E |Y|1|Y|>λ . λ→∞ n Taking λ → ∞, we get lim sup E |Xn |1|Xn |>λ = 0, λ→∞ n and hence {Xn } is uniformly integrable. s. and hence, in L1 ( ). 6), we define C = {B ∈ F : E(X∞ 1B ) = E(Y1B )} .

14) is called the Burkholder–Davis–Gundy inequality. It also holds for general p ≥ 1. Since in this book we will only need the case of p ≥ 2, whose proof is much easier than other cases, we will only state this case in the next theorem. 12 (Burkholder–Davis–Gundy inequality) Suppose that p ≥ 2 and X ∈ M2,c satisfying X0 = 0. Then there exists a constant Kp such that E max |Xs |p ≤ Kp E s≤t X p 2 t . 15) 45 46 3 : Stochastic integrals and Itô’s formula Proof Since |x|p ∈ C 2 for p ≥ 2, it follows from Itô’s formula that |Xt |p = t 0 p|Xs |p−2 Xs dXs + p(p − 1) 2 t 0 |Xs |p−2 d X s .

This point of view will be useful in Chapter 11. 11 Suppose that M ∈ Mloc . Let 0 = t0n < t1n < · · · < tnn = t be such that n max (tjn − tj−1 ) → 0. 3 Itô’s formula Then, n 2 n ) = M . (Mtjn − Mtj−1 t lim n→∞ j=1 Proof Note that n 2 n ) (Mtjn − Mtj−1 j=1 n tjn = n tj−1 j=1 =2 t 0 n )dMs + M 2(Ms − Mtj−1 tjn − M n tj−1 n Ms dMs − 2 n (Mt n − Mt n ) + M Mtj−1 j j−1 t j=1 → M t, where the first step follows from Itô’s formula, and the last step follows from the definition of the stochastic integral.

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