Mathematical Modeling in Finance with Stochastic Processes
3.2. RUIN PROBABILITIES 101 This says that if the working capital is much greater than the amount required for victory, then the ...
102 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES game P[Tt+1=x+ 1|Tt=x] = p P[Tt+1=x− 1 |Tt=x] = q P[Tt+1=y|Tt=x] = 0 ...
3.2. RUIN PROBABILITIES 103 In a random walk starting at the origin find the probability that the pointa >0 will be reached ...
104 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES Prove: In a random walk starting ata >0 the probability to reach ...
3.2. RUIN PROBABILITIES 105 undervalued. Reasons for buy-backs include putting unused cash to use, raising earnings per share, i ...
106 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES University of California, San Diego, Department of Mathematics, A.M ...
3.3. DURATION OF THE GAMBLER’S RUIN 107 in the gambler’s ruin, just as for the probability of ruin or victory. Vocabulary Expec ...
108 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES is DT 0 = T 0 q−p − a q−p 1 −(q/p)T^0 1 −(q/p)a for p 6 =q and T 0 ( ...
3.3. DURATION OF THE GAMBLER’S RUIN 109 It follows that the general solution of the duration equation is: DT 0 =T 0 /(q−p) +A+B( ...
110 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES Proof that the duration is finite The following discussion of finite ...
3.3. DURATION OF THE GAMBLER’S RUIN 111 Problems to Work for Understanding (a) Using a trial function of the formDpT 0 =C+DT 0 ...
112 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES This problem is adapted fromStochastic Calculus and Financial Ap- p ...
3.4 A Stochastic Process Model of Cash Management to $18. (I don’t believe that there is any way to solve this problem using for ...
114 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES Section Starter Question Suppose that you have a stock of 5 units of ...
3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 115 The mathematical expressionδskis theKronecker delta δsk= { 1 ifk=s 0 if ...
116 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES Modeling We assume for a simple model that a bank’s cash level fluct ...
3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 117 3.3 Several typical cycles in a model of the reserve requirement. where ( ...
118 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES expected duration equation(which is +1) arises because the game will ...
3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 119 We can write this asWskp =−2 max(s−k,0) Then solving for the boundary con ...
120 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES nto find the average cost Expected total cost in cyclei =K+E[Ri], bu ...
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