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Stochastic Calculus and Differential Equations for Physics and Finance.
Provides graduate students and practitioners in physics and economics with a better understanding of stochastic processes.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2013.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Abbreviations; Introduction; 1 Random variables and probability distributions; 1.1 Particle descriptions of partial differential equations; 1.2 Random variables and stochastic processes; 1.3 The n-point probability distributions; 1.4 Simple averages and scaling; 1.5 Pair correlations and 2-point densities; 1.6 Conditional probability densities; 1.7 Statistical ensembles and time series; 1.8 When are pair correlations enough to identify a stochastic process?; Additional reading; Exercises; 2 Martingales, Markov, and nonstationarity; 2.1 Statistically independent increments.
- 2.2 Stationary increments2.3 Martingales; 2.4 Nonstationary increment processes; 2.5 Markov processes; 2.6 Drift plus noise; 2.7 Gaussian processes; 2.8 Stationary vs. nonstationary processes; Additional reading; Exercises; 3 Stochastic calculus; 3.1 The Wiener process; 3.2 Ito's theorem; 3.3 Ito's lemma; 3.4 Martingales for greenhorns; 3.5 First-passage times; Additional reading; Exercises; 4 Ito processes and Fokker-Planck equations; 4.1 Stochastic differential equations; 4.2 Ito's lemma; 4.3 The Fokker-Planck pde; 4.4 The Chapman-Kolmogorov equation; 4.5 Calculating averages.
- 4.6 Statistical equilibrium4.7 An ergodic stationary process; 4.8 Early models in statistical physics and finance; 4.9 Nonstationary increments revisited; Additional reading; Exercises; 5 Selfsimilar Ito processes; 5.1 Selfsimilar stochastic processes; 5.2 Scaling in diffusion; 5.3 Superficially nonlinear diffusion; 5.4 Is there an approach to scaling?; 5.5 Multiaffine scaling; Additional reading; Exercises; 6 Fractional Brownian motion; 6.1 Introduction; 6.2 Fractional Brownian motion; 6.3 The distribution of fractional Brownian motion; 6.4 Infinite memory processes.
- 6.5 The minimal description of dynamics6.6 Pair correlations cannot scale; 6.7 Semimartingales; Additional reading; Exercises; 7 Kolmogorov's pdes and Chapman-Kolmogorov; 7.1 The meaning of Kolmogorov's first pde; 7.2 An example of backward-time diffusion; 7.3 Deriving the Chapman-Kolmogorov equation for an Ito process; Additional reading; Exercise; 8 Non-Markov Ito processes; 8.1 Finite memory Ito processes?; 8.2 A Gaussian Ito process with 1-state memory; 8.3 McKean's examples; 8.4 The Chapman-Kolmogorov equation; 8.5 Interacting system with a phase transition.
- 8.6 The meaning of the Chapman-Kolmogorov equationAdditional reading; Exercise; 9 Black-Scholes, martingales, and Feynman-Kac; 9.1 Local approximation to sdes; 9.2 Transition densities via path integrals; 9.3 Black-Scholes-type pdes; Additional reading; Exercise; 10 Stochastic calculus with martingales; 10.1 Introduction; 10.2 Integration by parts; 10.3 An exponential martingale; 10.4 Girsanov's theorem; 10.5 An application of Girsanov's theorem; 10.6 Topological inequivalence of martingales with Wiener processes; 10.7 Solving diffusive pdes by running an Ito process; 10.8 First-passage times.