Mathematical Modeling in Finance with Stochastic Processes
5.2 The Definition of Brownian Motion and the Wiener Process Problems to Work for Understanding Consider a random walk with a s ...
162 CHAPTER 5. BROWNIAN MOTION Section Starter Question Some mathematical objects are defined by a formula or an expression. Som ...
5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 163 Paris stock exchange, and so is the originator of the idea of ...
164 CHAPTER 5. BROWNIAN MOTION This is a statement of theMarkov propertyof the Wiener process. Recall that the sum of independen ...
5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 165 5.1 Graph of the Dow-Jones Industrial Average from August, inc ...
166 CHAPTER 5. BROWNIAN MOTION agreement. The Central Limit Theorem provides a reason to believe the agreement, assuming the req ...
5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 167 on a scale of day by day or week by week so the short-time cha ...
168 CHAPTER 5. BROWNIAN MOTION whereK 1 ,K 2 , andK 3 are constants that do not depend onx. For example, K 1 is the product of 1 ...
5.2. THE DEFINITION OF BROWNIAN MOTION AND THE WIENER PROCESS 169 (a) Find the probability that 0< W(1)<1. (b) Find the pr ...
170 CHAPTER 5. BROWNIAN MOTION For two random variablesXandY, statisticians call Cov(X,Y) =E[(X−E[X])(Y−E[Y])] the covariance o ...
5.3 Approximation of Brownian Motion by Coin-Flipping Sums 5.3 Approximation of Brownian Motion by Coin-Flipping Sums Rating Mat ...
172 CHAPTER 5. BROWNIAN MOTION be a sequence of independent, identically distributed Bernoulli random vari- ables. Note that Var ...
5.3. APPROXIMATION OF BROWNIAN MOTION BY COIN-FLIPPING SUMS 173 Theorem 1 .The joint distributions ofWˆN(t) converges to the joi ...
174 CHAPTER 5. BROWNIAN MOTION 5.4 Transformations of the Wiener Process Rating Mathematically Mature: may contain mathematics b ...
5.7 Quadratic Variation of the Wiener Process Mathematical Ideas Transformations of the Wiener Process A set of transformations ...
176 CHAPTER 5. BROWNIAN MOTION (c) Wshift(0) =W(0 +h)−W(h) = 0. (d) As the composition and difference of continuous functions,Ws ...
5.4. TRANSFORMATIONS OF THE WIENER PROCESS 177 Proof. To show that Winv(t) =tW(1/t) is a Wiener process by the four definitional ...
178 CHAPTER 5. BROWNIAN MOTION lemma and is beyond the scope of this course. Use the translation property in the third statement ...
5.5 Hitting Times and Ruin Probabilities Yuval Peres, University of California Berkeley, Department of Statis- tics Notes on sa ...
180 CHAPTER 5. BROWNIAN MOTION Mathematical Ideas Hitting Times Consider the standard Wiener processW(t), which starts atW(0) = ...
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