Stochastic Calculus and Stochastic Filtering lecture notes
File : pdf, 488 KB, 95 pages
by Alan Bain
Contents
1. Introduction
2. Contents
3. Stochastic Processes
3.1. Probability Space
3.2. Stochastic Process
4. Martingales
4.1. Stopping Times
5. Basics
5.1. Local Martingales
5.2. Local Martingales which are not Martingales
6. Total Variation and the Stieltjes Integral
6.1. Why we need a Stochastic Integral
6.2. Previsibility
6.3. Lebesgue-Stieltjes Integral
7. The Integral
7.1. Elementary Processes
7.2. Strictly Simple and Simple Processes
8. The Stochastic Integral
8.1. Integral for H in L and M in M_2
8.2. Quadratic Variation
8.3. Covariation
8.4. Extension of the Integral to L^2(M)
8.5. Localisation
8.6. Some Important Results
9. Semimartingales
10. Relations to Sums
10.1. The UCP topology
10.2. Approximation via Riemann Sums
11. Ito’s Formula
11.1. Applications of Ito’s Formula
11.2. Exponential Martingales
12. Levy Characterisation of Brownian Motion
13. Time Change of Brownian Motion
13.1. Gaussian Martingales
14. Girsanov’s Theorem
14.1. Change of measure
15. Brownian Martingale Representation Theorem
16. Stochastic Differential Equations
17. Relations to Second Order PDEs
17.1. Infinitesimal Generator
17.2. The Dirichlet Problem
17.3. The Cauchy Problem
17.4. Feynman-Kac Representation
18. Stochastic Filtering
18.1. Signal Process
18.2. Observation Process
18.3. The Filtering Problem
18.4. Change of Measure
18.5. The Unnormalised Conditional Distribution
18.6. The Zakai Equation
18.7. Kushner-Stratonowich Equation
19. Gronwall’s Inequality
20. Kalman Filter
20.1. Conditional Mean
20.2. Conditional Covariance
21. Discontinuous Stochastic Calculus
21.1. Compensators
21.2. RCLL processes revisited
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