Mathematical Modeling in Finance with Stochastic Processes
1.5. MATHEMATICAL MODELING 41 Modeling A good description of the model will begin with an organized and complete description of ...
42 CHAPTER 1. BACKGROUND IDEAS computer program. Programs written in either high-level languages such as C, FORTRAN or Basic and ...
1.5. MATHEMATICAL MODELING 43 optima or the rates of change of optima with respect to the constraints. If the model is a differe ...
44 CHAPTER 1. BACKGROUND IDEAS for the model. On the other hand, if a predicted or modeled value varies substantially in compari ...
1.5. MATHEMATICAL MODELING 45 automatically or naturally. The craft of creating, solving, using, and in- terpreting a mathematic ...
46 CHAPTER 1. BACKGROUND IDEAS From this limited set of assumptions about theoretical entities called molecules physicists can d ...
1.5. MATHEMATICAL MODELING 47 The additional constantsaandbrepresent the new elements of intermolecular attraction and volume ef ...
48 CHAPTER 1. BACKGROUND IDEAS is intrinsically unobtainable.” If the dollar amounts get very large (so that rationality no long ...
1.6. RANDOMNESS 49 model that summarizes our experience with many coins. In the context of statistics, this is called thefrequen ...
50 CHAPTER 1. BACKGROUND IDEAS We assign the probability 1/2 to the event that the coin will land heads and probability to 1/2 t ...
1.6. RANDOMNESS 51 locity and rotational velocity lead to different outcomes. The assignment of probabilities 1/2 to heads and t ...
52 CHAPTER 1. BACKGROUND IDEAS 1.5 Initial conditions for a coin flip, from Keller ...
1.6. RANDOMNESS 53 1.6 Persi Diaconis’ mechanical coin flipper Randomness and the Markets A branch of financial analysis, genera ...
54 CHAPTER 1. BACKGROUND IDEAS the actions of thousands of people. The economic principles at work on the variables are understo ...
1.7 Stochastic Processes cific N-13 atom to a C-13 isotope is apparently truly random, since it seems we fundamentally cannot de ...
56 CHAPTER 1. BACKGROUND IDEAS Key Concepts A sequence or interval of random outcomes, that is to say, a string of random outco ...
1.7. STOCHASTIC PROCESSES 57 A generalization of a Markov chain is aMarkov Process. In a Markov process, we allow the index set ...
58 CHAPTER 1. BACKGROUND IDEAS then we usually write the index variable or time variable as a subscript. ThusXnwould be the usua ...
1.7. STOCHASTIC PROCESSES 59 The defining property of a Markov chain is that P[Xj=l|X 0 =k 0 ,X 1 =k 1 ,...,Xj− 1 =kj− 1 ] =P[Xj ...
60 CHAPTER 1. BACKGROUND IDEAS up to timetby using a Geiger counter. The discrete state variable is the counting number of click ...
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