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 process applications B(Rd Banach space Bb(Rd Borel measurable bounded Brownian motion cadlag called Chapter characteristic function closable Co(Rd compound Poisson process continuous Corollary deduce define denote Dirichlet forms distribution domain Example Exercise exists exponential Feller process filtration finite variation Gaussian generalisation given Hence independent inequality infinitely divisible interlacing Ito's formula jumps Kunita Lemma Levy measure Levy process Levy symbol Levy-Ito decomposition Levy-type stochastic integral linear operator Lipschitz mapping Markov process martingale martingale-valued measure matrix Note obtain oo a.s. Poisson random measure positive definite probability measure probability space proof of Theorem Proposition random variable required result follows S(Rd satisfies Sato SDEs Section self-adjoint semigroup semimartingales sequence standard Brownian motion stochastic integrals stochastic process stock prices subordinator Subsection symmetric taking values u e Rd vector write x e Rd xN(t,dx