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Extra resources for Algebraic Structure Theory of Sequential Machines [appl math]

Sample text

A decomposition in which the process a (t) is natural is unique. Proof. Let ~(t) be a supermartingale of dass DL. Then supermartingale of dass D, and in view of Theorem 9 ~a (t) = ~(a " t), a > 0, is a where /-ta(t) is a uniformly integrable martingale, and aa(t) is an integrable natural process. Let b > a. Then ~a (t) = ~b (t " a) = /-tb (t " a ) - ab (t " a), and it follows from the uniqueness of Doob's decomposition that /-tb (t) = /-ta (t) and ab (t) = aa (t) for t ~ a. im /-ta (t) and a (t) = lim aa (t) exist with prob ability 1 and, moreover, /-t (t) is dearly a martingale, while a (t) is a natural process and ~(t) = /-t (t) - a (t).

A decomposition in which the process a (t) is natural is unique. Proof. Let ~(t) be a supermartingale of dass DL. Then supermartingale of dass D, and in view of Theorem 9 ~a (t) = ~(a " t), a > 0, is a where /-ta(t) is a uniformly integrable martingale, and aa(t) is an integrable natural process. Let b > a. Then ~a (t) = ~b (t " a) = /-tb (t " a ) - ab (t " a), and it follows from the uniqueness of Doob's decomposition that /-tb (t) = /-ta (t) and ab (t) = aa (t) for t ~ a. im /-ta (t) and a (t) = lim aa (t) exist with prob ability 1 and, moreover, /-t (t) is dearly a martingale, while a (t) is a natural process and ~(t) = /-t (t) - a (t).

The quadratic variation of process (t) on the interval [0, Tl is denoted by = P-limA_o lT~ (t). [(, n, [(, n 42 I. Martingales and Stochastic Integrals Remark. If the sampie functions of a process (t) are continuous and of bounded variation on [0, T) with probability 1, then [~, ~ 1r = o. C(td- (tk-l)! is the variation of (I) on [0, T). (td- (tk-l)! ~ 0 as A ~ 0, it follows that CT~ ~ 0 with probability 1. 0 Lemma 8. 11 ~(t) is a square integrable martingale on [0, T] then the lamily 01 variables {CT~ (T)} is unilormly integrable.