Mathematical Modeling in Finance with Stochastic Processes
3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 121 That is, the optimal value of the maximum amount of cash to keep varies a ...
122 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES using the trial function Wskp = { C+Ds ifs≤k E+Fs ifs > k. Show ...
Chapter 4 Limit Theorems for Stochastic Processes 4.1 Laws of Large Numbers Rating Mathematically Mature: may contain mathematic ...
4 Limit Theorems for Stochastic Processes Vocabulary TheWeak Law of Large Numbersis a precise mathematical state- ment of what ...
4.1. LAWS OF LARGE NUMBERS 125 with probability densityf: E[X] = ∫∞ 0 xf(x)dx = ∫a 0 xf(x)dx+ ∫∞ a xf(x)dx ≥ ∫∞ a xf(x)dx ≥ ∫∞ a ...
126 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Proof.Since the mean of a sum of random variables is the sum of the means ...
4.1. LAWS OF LARGE NUMBERS 127 The proof of this theorem is beautiful and deep, but would take us too far afield to prove it. Th ...
128 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Sources This section is adapted from Chapter 8, “Limit Theorems”,A First ...
4.2 Moment Generating Functions Section Starter Question Give some examples of transform methods in mathematics, science or engi ...
130 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Xi,E[Xi],P[Xi] −→ φX(t) y conclusions ←− calculations 4.1 Block diagra ...
4.2. MOMENT GENERATING FUNCTIONS 131 for all valuestfor which the integral converges. Example.Thedegenerate probability distribu ...
132 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Then φ′′X(0) =E [ X^2 ] . Continuing in this way: φ(Xn)(0) =E[Xn] In word ...
4.2. MOMENT GENERATING FUNCTIONS 133 Proof. φZ(t) =E [ etX ] = 1 √ 2 πσ^2 ∫∞ −∞ etxe−(x−μ) (^2) /(2σ (^2) ) dx = 1 √ 2 πσ^2 ∫∞ − ...
134 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES An alternative visual proof that the sum of independent normal random var ...
4.3 The Central Limit Theorem Key Concepts The statement, meaning and proof of the Central Limit Theorem. We expect the normal ...
136 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Likewise, convergence of probabilities of events implies convergence in d ...
4.3. THE CENTRAL LIMIT THEOREM 137 wherer 2 (t/n) is a error term such that lim n→∞ r(t/n) (1/n) = 0. Then we need to consider φ ...
138 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Remark.The Central Limit Theorem is true even under the slightly weaker a ...
4.3. THE CENTRAL LIMIT THEOREM 139 The first version of the central limit theorem was proved by Abraham de Moivre around 1733 fo ...
140 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES Figure 4.2: Approximation of the binomial distribution with the normal di ...
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