LÚvy Processes and Stochastic Calculus
LÚvy 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 LÚvy processes. The second part accessibly develops the stochastic calculus for LÚvy processes. All the tools needed for the stochastic approach to option pricing, including It˘'s formula, Girsanov's theorem and the martingale representation theorem are described.
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adapted applications argument associated assume backwards Banach space bounded Brownian motion called Chapter characteristic clearly closed condition consider construction continuous convergence deduce define definite denote differential equations diffusion Dirichlet distribution establish Example Exercise exists extension fact Feller filtration finite flow follows formula function Gaussian generalisation give given Hence important independent inequality infinitely divisible interesting introduce It˘ jumps Lemma LÚvy process limit linear operator mapping Markov process martingale measure n e N Note obtain operator option paths Poisson process positive probability probability measure Proof Proposition random variable reader respect result result follows satisfies SDEs semigroup sequence solution space stable standard stochastic differential stochastic differential equations stochastic integral subordinator suppose symbol symmetric Theorem theory unique usual write x e Rd