By Fima C Klebaner
This ebook offers a concise remedy of stochastic calculus and its purposes. It supplies an easy yet rigorous therapy of the topic together with more than a few complicated issues, it's precious for practitioners who use complicated theoretical effects. It covers complicated purposes, comparable to types in mathematical finance, biology and engineering.
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Extra resources for Introduction to Stochastic Calculus with Applications
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We illustrate this on our financial example. 6: Take T = 3, and assume that at each trading time the stock can go up by the factor u or down by d. Ω= u u u d u u u u d d u d u d u d d u u d d d d d A ¯ B B ¯ A ¯ Look at the sets generated by information about S1 . This is a partition of Ω, {A, A}. Together with the empty set and the whole set, this is the field F1 . Sets generated ¯ Thus the sets formed by knowledge of S1 and by information about S2 are B and B. S2 is the partition of Ω, consisting of all intersections of the above sets.
CONCEPTS OF PROBABILITY THEORY 7. Monotone convergence. If 0 ≤ Xn , and Xn ↑ X with E|X| < ∞, then E Xn |G ↑ E X|G . 25) 8. Fatous’ lemma. If 0 ≤ Xn , then E lim inf Xn |G ≤ lim inf E Xn |G . 26) 9. Dominated convergence. If limn→∞ Xn = X almost surely and |Xn | ≤ Y with EY < ∞, then lim E Xn |G = E X|G . g. Breiman (1968), Chapter 4. The conditional probability P(A|G) is defined as the conditional expectation of the indicator function, P(A|G) = E(IA |G), and it is a G-measurable random variable, defined P-almost surely.
The following result, which is known as Slutskii theorem, is frequently used in applications. 15 If Xn converges to X and Yn converges to Y , then Xn + Yn converges to X + Y , for any type of stochastic convergence, except for convergence in distribution. However, if Y = 0 or Xn and Yn are independent, then the result is also true for convergence in distribution. 16 (Monotone convergence) If Xn ≥ 0, and Xn are increasing to a limit X, which may be infinite, then limn→∞ EXn = EX. 17 (Fatou’s lemma) If Xn ≥ 0 (or Xn ≥ c > −∞), then E(lim inf n Xn ) ≤ lim inf n EXn .