Mathematical Modeling in Finance with Stochastic Processes
4.3. THE CENTRAL LIMIT THEOREM 141 LetXi= 1 (success) if the two-child family is girl-girl, andXi= 0 (failure) if the two-child ...
142 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES ferings. Assume that each bank has a certain amount of funds available fo ...
4.3. THE CENTRAL LIMIT THEOREM 143 Sources The proofs in this section are adapted from Chapter 8, “Limit Theorems”, A First Cour ...
144 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES daynwithS 0 given, then Sn=Sn− 1 +Xn,n≥ 1 whereX 1 ,X 2 ,...are independe ...
4.4 The Absolute Excess of Heads over Tails distributionXwith a parameter 1/100 which implies thatE[X] = 100 and Var [X] = 100^2 ...
146 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Key Concepts The probability that the number of heads exceeds the number ...
4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 147 What is the probability of an excess of a fixed number of heads over tails or ...
148 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES 4.3 The half-integer correction ...
4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 149 the difference between the number of heads and tails, an excess of heads if pos ...
150 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES asn→∞. RewritingP [∣∣Tn n ∣ ∣> ]=P[|Tn|> n] this restatement of t ...
4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 151 4.4 Probability ofsexcess heads in 500 tosses Illustration 2 What is the probab ...
152 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES vague question precise by asking “How good does the information from the ...
4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 153 0 .07 to calculate thatP[X 1 ]< 0 ≈ 0 .47210, or about 47%. Alternatively, i ...
154 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES (c) What is the approximate probability that the number of heads is withi ...
Chapter 5 Brownian Motion 5.1 Intuitive Introduction to Diffusions Rating Mathematically Mature: may contain mathematics beyond ...
156 CHAPTER 5. BROWNIAN MOTION Mathematical Ideas The question is “How should we set up the limiting process so that we can make ...
5.1 Intuitive Introduction to Diffusions random walk on the axis, we then need to know the size ofδTn. Now E[δ·Tn] = (p−q)·δ·n a ...
158 CHAPTER 5. BROWNIAN MOTION p+q = 1, sop → 1 /2. The analytical formulation of the problem is as follows. Letδbe the size of ...
5.1. INTUITIVE INTRODUCTION TO DIFFUSIONS 159 Sincevk,nis the probability of findingTn·δbetweenk·δand (k+2)·δ, and since this in ...
160 CHAPTER 5. BROWNIAN MOTION In the limit ask→∞andn→∞,vk,nwill be the sampling of the function v(t,x) at time intervalsr, so t ...
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