Brownian motion : an introduction to stochastic processes /
René L. Schilling, Lothar Partzsch ; with a chapter on simulation by Björn Böttcher.
Bok Engelsk 2012 · Electronic books.
| Utgitt | Berlin ; Boston : : De Gruyter, , c2012.
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| Omfang | 1 online resource (396 p.)
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| Opplysninger | Description based upon print version of record.. - Preface; Dependence chart; Index of notation; 1 Robert Brown's new thing; 2 Brownian motion as a Gaussian process; 2.1 The finite dimensional distributions; 2.2 Invariance properties of Brownian motion; 2.3 Brownian Motion in Rd; 3 Constructions of Brownian motion; 3.1 The Lévy-Ciesielski construction; 3.2 Lévy's original argument; 3.3 Wiener's construction; 3.4 Donsker's construction; 3.5 The Bachelier-Kolmogorov point of view; 4 The canonical model; 4.1 Wiener measure; 4.2 Kolmogorov's construction; 5 Brownian motion as a martingale; 5.1 Some 'Brownian' martingales. - 16.4 Itô's formula for stochastic differentials16.5 Itô's formula for Brownian motion in Rd; 16.6 Tanaka's formula and local time; 17 Applications of Itô's formula; 17.1 Doléans-Dade exponentials; 17.2 Lévy's characterization of Brownian motion; 17.3 Girsanov's theorem; 17.4 Martingale representation - 1; 17.5 Martingale representation - 2; 17.6 Martingales as time-changed Brownian motion; 17.7 Burkholder-Davis-Gundy inequalities; 18 Stochastic differential equations; 18.1 The heuristics of SDEs; 18.2 Some examples; 18.3 Existence and uniqueness of solutions. - 18.4 Solutions as Markov processes. - 5.2 Stopping and sampling5.3 The exponential Wald identity; 6 Brownian motion as a Markov process; 6.1 The Markov property; 6.2 The strong Markov property; 6.3 Desiré André's reflection principle; 6.4 Transience and recurrence; 6.5 Lévy's triple law; 6.6 An arc-sine law; 6.7 Some measurability issues; 7 Brownian motion and transition semigroups; 7.1 The semigroup; 7.2 The generator; 7.3 The resolvent; 7.4 The Hille-Yosida theorem and positivity; 7.5 Dynkin's characteristic operator; 8 The PDE connection; 8.1 The heat equation; 8.2 The inhomogeneous initial value problem. - 8.3 The Feynman-Kac formula8.4 The Dirichlet problem; 9 The variation of Brownian paths; 9.1 The quadratic variation; 9.2 Almost sure convergence of the variation sums; 9.3 Almost sure divergence of the variation sums; 9.4 Levy's characterization of Brownian motion; 10 Regularity of Brownian paths; 10.1 Hölder continuity; 10.2 Non-differentiability; 10.3 Lévy's modulus of continuity; 11 The growth of Brownian paths; 11.1 Khintchine's Law of the Iterated Logarithm; 11.2 Chung's 'other' Law of the Iterated Logarithm; 12 Strassen's Functional Law of the Iterated Logarithm. - Stochastic processes occur in a large number of fields in sciences and engineering, so they need to be understood by applied mathematicians, engineers and scientists alike. This work is ideal for a first course introducing the reader gently to the subject matter of stochastic processes. It uses Brownian motion since this is a stochastic process which is central to many applications and which allows for a treatment without too many technicalities. All chapters are modular and are written in a style where the lecturer can ""pick and mix"" topics. A ""dependence chart"" will guide the reader when
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| ISBN | 3540645543
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