Infinite-dimensional Analysis: Operators In Hilbert Space; Stochastic Calculus Via Representations, And Duality Theory by Palle Jorgensen;James Tian;

Author:Palle Jorgensen;James Tian;
Language: eng
Format: epub
ISBN: 9789811225796
Publisher: World Scientific Publishing Company
Published: 2021-06-15T00:00:00+00:00

where K (t, ·) is a measure for every t.

Assume K (t, ·) ≪ λ with Radon–Nikodym derivative

then

Further assume that K is compact and positive (selfadjoint), and let

be its spectral decomposition, i.e., Kφn = λnφn, and {φn} is an ONB in L2 (λ). Then

Lemma 4.40 (Karhunen–Loève). Let K be as in (4.46). Let {Zn} be i.i.d. random variables with Zn ~ N (0, 1). Set : then

Proof. We have

Example 4.41. The standard Brownian bridge in [0, π] can be represented as

where Zn (·) is i.i.d. and Zn ~ N (0,1). See Figure 4.2. Then

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