COMPG008 - Stochastic Processes for Finance
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Calculus, Basic Probability
Guido Germano (100%)
Rehearse/survey probability theory and give a systematic introduction to stochastic processes and their applications without stressing too much the measure-theoretical aspects and other mathematical formalisms. The main target are students with an undergraduate degree in engineering, physics, computer science and the like, who have a good basis in calculus and have already come into contact with aspects of probability and statistics for ad hoc applications like transport equations, laboratory data treatment, and quantum mechanics, but have not attended yet a dedicated course on stochastic processes. The course material will unfold with references to its historical development and early applications in physics/engineering the students may already heard of, ending with current-day applications in finance.
Familiarity with probability theory, stochastic processes in discrete and continuous time, stochastic calculus, and basic applications in physics/engineering and, mainly, in finance.
1. Probability Theory - a Survey (3 or 4 weeks)
Definitions, Kolmogorov's axioms
Joint and conditional probabilities, Bayes' theorem
Probability distribution and density functions
Multivariate and marginal distributions
Mean, variance, covariance, correlation, moments, cumulants
Moment/cumulant-generating functions, characteristic function, entropy
Sum and transformation of random variables
Gaussian and Poisson distributions
Law of large numbers and central limit theorem
2. Stochastic Processes (6 or 7 weeks)
Definitions and classification
Markov and semi-Markov processes, Chapman-Kolmogorov equation
Jump processes: the master equation
Diffusion processes: the Fokker-Planck equation
Deterministic processes: the Liouville equation
General processes; martingales and semimartingales
Evolution equations, forward and backward equations
Stationary and homogeneous Markov processes, ergodicity
Auto- and cross-covariance/correlation
Random walk, Brownian motion, Wiener process
Poisson process, compound Poisson process
Renewal process, continuous-time random walk
Stochastic integration: Ito and Stratonovich; Ito's formula
Langevin equation, stochastic differential equations, Feynman-Kac theorem
Euler-Maruyama and Milstein algorithms
Geometric Brownian motion and Black-Scholes equation
Ornstein-Uhlenbeck process and Vasicek model
Rayleigh and Bessel processes, Cox-Ingersoll-Ross model
Method of Instruction
30 hours of lectures plus homework and assignments.
The course has the following assessment components:
- Written exam (2.5 hours, 100%)
To pass this course, students must:
- Obtain a pass mark of 50%.
Crispin W. Gardiner, Stochastic Methods - A Handbook for the Natural and Social Sciences, 4th ed., Springer 2009, Chapters 1-4, 10, 15, plus a few other selected parts, for a total of 150-200 out of 420 pages. See table of contents on http://www.springer.com/physics/complexity/book/978-3-540-70712-7 and a review of the 3rd edition (2004) on http://www.ejwagenmakers.com/2006/GardinerBookReview2006.pdf.
- J. L. McCauley, Stochastic Calculus and Differential Equations for Physics and Finance, Cambridge University Press, 2013.
- N. van Kampen, Stochastic Processes, 3rd edition, Elsevier, 2007.
- W. Paul, J. Baschnagel, Stochastic Processes, Springer, 1999.
- Sheldon N. Ross, Stochastic Processes, 2nd edition, Wiley, 1996.
- S. Karlin, H. M. Taylor, An Introduction to Stochastic Modeling, 3rd edition, Academic Press, 1998.
- S. Karlin, H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, 1981.
- S. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975.