Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk TheoryProviding the necessary materials within a theoretical framework, this volume presents stochastic principles and processes, and related areas. Over 1000 exercises illustrate the concepts discussed, including modern approaches to sample paths and optimal stopping. |
Contents
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Definition of stochastic process Cylinder σalgebra
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1 |
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Characteristics of a stochastic process Mean and covariance
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11 |
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Trajectories Modifications Filtrations
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21 |
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Continuity Differentiability Integrability
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33 |
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and Poisson processes Poisson point measures
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43 |
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Gaussian processes
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59 |
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7
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68 |
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8
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103 |
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Hints
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152 |
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metrics Functional limit theorems
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241 |
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Statistics of stochastic processes
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271 |
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Stochastic processes in financial mathematics discrete time
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303 |
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Stochastic processes in financial mathematics continuous time 315
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314 |
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Basic functionals of the risk theory
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327 |
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Appendix
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359 |
| 371 | |
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arbitrage-free arbitrary Assume Black–Scholes Borel call option Chapter characteristic function Consider converges corresponding covariance function Definition Denote density distribution function dX(t EX(t exists filtration Find finite finite-dimensional distributions function f Gaussian process holds true i.i.d. random variables independent increments inequality inf{t Itó formula lemma Let X(t Let Xn Markov chain Markov process Markov property matrix mean square metric moment generating function nonnegatively defined O-algebra orthogonal p_(s parameter phase space Poisson process previous problem probability space process with independent process with intensity Prove queue random element random function random variables random walk request respect satisfies solution Springer Science+Business Media statement stochastic differential stochastic integral stochastic process stochastic process X(t submartingale supermartingale t e Rt Theorem trajectories transition function transition probabilities uniformly integrable vector Wasserstein metric wide-sense stationary Wiener process zero