## Lévy Processes and Stochastic CalculusLévy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. For the first time in a book, Applebaum ties the two subjects together. He begins with an introduction to the general theory of Lévy processes. The second part accessibly develops the stochastic calculus for Lévy processes. All the tools needed for the stochastic approach to option pricing, including Itô's formula, Girsanov's theorem and the martingale representation theorem are described. |

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### Contents

Martingales stopping times and random measures | 90 |

Markov processes semigroups and generators | 120 |

Stochastic integration | 190 |

Exponential martingales change of measure and financial | 246 |

Stochastic differential equations | 292 |

References | 360 |

375 | |

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adapted applications associated backwards Banach space Borel bounded Brownian motion called Chapter characteristic closed condition consider construction continuous convergence deduce define definite denote differential equations diffusion Dirichlet distribution domain establish Example Exercise exists extended fact Feller filtration finite flow follows formula function Gaussian generalisation give given Hence important independent inequality infinitely divisible interesting introduce Itô jumps Lemma Lévy process limit linear operator mapping Markov process martingale n e N natural Note obtain operator option paths Poisson process positive probability measure Proof Proposition random variable reader respect result result follows satisfies SDEs semigroup sequence solution sometimes space stable standard stochastic differential stochastic differential equations stochastic integrals subordinator suppose symmetric Theorem theory unique usual write x e Rd

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