Mathematical Modeling in Finance with Stochastic Processes
5.5. HITTING TIMES AND RUIN PROBABILITIES 181 path, it varies withω. On the other hand, the shifting transformation defined in t ...
182 CHAPTER 5. BROWNIAN MOTION probability that the random walk goes up to valueabefore going down to valuebwhen the step size i ...
5.6. PATH PROPERTIES OF BROWNIAN MOTION 183 Suppose you own one share of stock whose price changes according to a Wiener proces ...
184 CHAPTER 5. BROWNIAN MOTION Vocabulary In probability theory, the termalmost surelyis used to indicate an event which occurs ...
5.6. PATH PROPERTIES OF BROWNIAN MOTION 185 equal. Even more to the point, the functiont^2 /^3 is continuous but not differ- ent ...
186 CHAPTER 5. BROWNIAN MOTION Theorem 23.With probability 1 (i.e. almost surely) the graph of a Brownian Motion path has Hausdo ...
5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 187 Section Starter Question What is an example of a function that “varies a lot” ...
188 CHAPTER 5. BROWNIAN MOTION Mathematical Ideas Variation Definition.A functionf(x) is said to havebounded variationif, over t ...
5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 189 Quadratic Variation of the Wiener Process We can guess that the Wiener Proces ...
190 CHAPTER 5. BROWNIAN MOTION by property 1 of the definition of standard Brownian motion. A routine (but omitted) computation ...
5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 191 Now let Znk= ( W (kt n ) −W ( (k−1)t n )) √ t/n Then for eachn, the sequenceZ ...
192 CHAPTER 5. BROWNIAN MOTION surprising given the law of the iterated logarithm. We know that in any neighborhood [t,t+dt] to ...
5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 193 Problems to Work for Understanding Show that a monotone increasing function ...
194 CHAPTER 5. BROWNIAN MOTION ...
Chapter 6 Stochastic Calculus 6.1 Stochastic Differential Equations and the Euler-Maruyama Method Rating Mathematically Mature: ...
196 CHAPTER 6. STOCHASTIC CALCULUS nents. The goal is to unravel the relation to find the stochastic process. Under mild conditi ...
6.1 Stochastic Differential Equations and the Euler-Maruyama Method processes locally from our base deterministic function, the ...
198 CHAPTER 6. STOCHASTIC CALCULUS Example.The next simplest stochastic differential equation is dX=σdW, X(0) =b This stochastic ...
6.1. STOCHASTIC DIFFERENTIAL EQUATIONS AND THE EULER-MARUYAMA METHOD 199 Stochastic Differential Equations: Numerically The samp ...
200 CHAPTER 6. STOCHASTIC CALCULUS Of course, this can be programmed and the step size made much smaller, presumably with better ...
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