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#2, Exam Math 401, Dr. M. Bohner, Dec 1, 2006. Name: There are 50 boxes. Please only answer 40 of them. Clearly indicate by crossing the box on the two diagonals which boxes you remove. If there are no (or not enough) boxes removed, I will remove the rst ten (or the remaining) boxes from above on the rst page. Let ( , F, F, P) be a ltered probability space. A random variable : N0 is called a stopping time if time if a . We say that a set A is determined by , and the set of all such sets determined by time is . Doob s optional sampling theorem says that if , are bounded stopping . Doob s stopping time principle . times with and X a martingale, then says that if is a bounded stopping time and X a martingale, then Let X be an adapted sequence of integrable random variables. The sequence Z de ned by ZN = and Zn = for 0 n N 1 is called the Snell ensupermartingale X. = velope of X. It is the solves the optimal stopping problem for X. The main characterization (not de nition) says that a stopping time is optimal for X i der to nd the largest stopping time, the . In ordecomposition is helpful. If G is an Vk = American derivative security, then we de ne its value process by for 0 k N 1, in particular is the Snell envelope of a ican V0 = Then . it turns out that in fact . Moreover, the discounted wealth process is then under the risk-neutral probability. We also showed that an Ameris equivalent to a European . Now continuous stochastic pro- cesses: A process X adapted to a ltration F is called a martingale if a.s. for all , W has and W has . The ve de ning properties of Brownian motion W say , , and increments, E(W (t)) = , . If W is Brownian motion, then we have V(W (t)) = , and Cov(W (s), W (t)) = W = . L vy s theorem says that the quadratic e . Geometric Brownian motion S is de. Furthersays that the series variation of Brownian motion is ned by more, log S t , and with = 0 it is a = . The theorem of L vy e t W (t) = n=0 de nes Brownian motion, where Zn are , n are given by , where n = 2j + k uniquely with integers j, k, and sn functions, that are constructed using the funcintegral I is de- are the n 1 tions. If X = i=0 Ci [ti ,ti+1 ) is a t process, then the ned by I(t) = 0 X(u)dW (u) = for tk t < tk+1 . and the It isometry says that o I = . The zero mean property says that . The quadratic variation of I is given by T t If W is Brownian motion, then 0 W (u)dW (u) = , and the It foro mula for Brownian motion says . An It process Y is de ned by o Y (t) = , t and 0 (u)dY (u) = .

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