## Essentials of Stochastic ProcessesThis test is designed for a Master's Level course in stochastic processes. It features the introduction and use of martingales, which allow one to do much more with Brownian motion, e.g., option pricing, and queueing theory is integrated into the Continuous Time Markov Chain and Renewal Theory chapters as examples. |

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1 | |

2 Poisson Processes | 95 |

3 Renewal Processes | 125 |

4 Continuous Time Markov Chains | 147 |

5 Martingales | 201 |

6 Mathematical Finance | 223 |

A Review of Probability | 251 |

268 | |

271 | |

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balls binomial model Black–Scholes bulb call option cars chain with transition compute concrete example Consider Customers arrive define definition density function desired result detailed balance condition distribution with mean distribution with rate equation expected number expected value exponential amount exponentially distributed amount Find the stationary finite formula Formulate a Markov geometric distribution gives greatest common divisor hour implies irreducible irreducible set jump large numbers law of large Lemma Let Xn limiting fraction long run machine Markov chain Markov property martingale minfn minutes number of customers outcomes Poisson distribution Poisson process positive recurrent process with rate Proof queue random variables random walk renewal process replaced risk neutral probability satisfies the detailed server solve space stationary distribution step stock price supermartingale Suppose transient transition matrix transition probability variance vector waiting XnC1