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. |
Contents
1 | 11 |
Martingales stopping times and random measures | 70 |
2 | 78 |
The jumps of a Lévy process Poisson random measures | 86 |
5 | 111 |
8 | 117 |
Stochastic integration | 190 |
Exponential martingales change of measure and financial | 246 |
Stochastic differential equations | 292 |
References | 360 |
Index of notation | 375 |
Common terms and phrases
adapted process B(Rd backwards Borel measurable bounded Brownian motion càdlàg called Chapter characteristic function closable Co(Rd compound Poisson process condition continuous deduce define denote diffusion Dirichlet form distribution E(ei Example Exercise exists exponential Feller process filtration generalisation given Hence independent inequality infinitely divisible interlacing Itô Itô's formula jumps Kunita L²(Rd Lemma Lévy measure Lévy process Lévy symbol Lévy-Itô decomposition Lévy-type stochastic integral linear operator Lipschitz mapping Markov process martingale mathematical Note o-algebra obtain Poisson random measure positive definite Prob probability measure probability space Proposition pseudo-differential operator random variable required result follows S(Rd sample paths Sato SDEs Section self-adjoint semigroup semimartingales sequence solution Springer-Verlag standard Brownian motion Stoch stochastic calculus stochastic differential equations stochastic integral stochastic process stock prices subordinator Subsection symmetric T₁ theory tj+1 unique y₁