SCOTTISH MATHEMATICAL SCIENCES TRAINING CENTRE - PROBABILITY STREAM

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NOTES

1. The proposed syllabus is intended to provide the core skills in
   probability necessary for students undertaking research in
   probability and statistics.  With regard to the latter, it probably
   goes considerably beyond what is needed.  The course is also
   intended to be accessible to students in other areas of
   mathematics.

2. Although most students may be expected to have had some prior
   training in at least elementary probability, the level of such
   skill is likely to be extremely variable.  Hence the rapid revision
   corresponding to the first 4-5 lectures below is felt to be
   essential.  The treatment of this material is likely to contain a
   certain amount that is new to everybody.

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PROPOSED SYLLABUS


FOUNDATIONS

1.  Probability spaces.							SZa
      Events, probability measures, random variables: definitions,
      axioms, basic results, elementary calculations, examples, simple
      combinatorics.

2.  Conditioning and independence.					SZa
      Conditional probability, chain rule, partition theorem (law of
      total probability). Independence of events and random variables.
      Bernoulli schemes.

3.  Random variables and their distributions.				TK
      Distribution functions, probability (mass) functions,
      probability density functions, moments, simple inequalities
      (Markov, Chebychev), transformations.  Moment generating
      functions (briefly).

4.  Joint distributions and independence.				TK
      Joint distributions, marginal and conditional
      distributions. Independence: factorisation of joint distribution
      and moment generating function, moment properties.
      Covariance, correlation, multivariate normal distribution.

5.  Important special distributions.					TK
      Binomial, Poisson, geometric, uniform, exponential and normal
      distributions: definitions, properties and the relations between
      them.

6.  Convergence and related concepts.					TC
      Convergence in distribution, in probability, in $L^p$,
      a.s. convergence, and relations between them.  Uniform
      integrability.  Borel-Cantelli lemmas.  Weak law of large
      numbers.

7.  Sequences of independent random variables.				TC
      0-1 law  Strong law of large numbers (proof for finite fourth
      moment case only).  Central limit theorem.  Possible extensions
      to non-independent sequences.
 

CONDITIONAL EXPECTATION AND MARTINGALES (DISCRETE PARAMETER SPACE)

8.  Conditional expectation and martingales.				AD/IG
      Conditional expectation definition, tower property.  Martingales
      (including sub and super): definitions, optional stopping
      theorems and applications.

9.  Martingale limit theorems and applications.				AD/IG
      (Proofs for $L^2$ theory only.)   Ruin problems.


MARKOV PROCESSES (COUNTABLE STATE SPACE)

10. Markov chains in discrete time.					XM/SZ
      Definitions, transition matrix, evolution of probability measure
      and simple calculations.  Strong Markov property.  Irreducible
      chains: existence and uniqueness of stationary distributions,
      relation to recurrence and transience properties detailed
      balance, limiting behaviour, ergodic theorems.

11. Poisson processes on the real line.					XM/SZu
      Homogeneous processes (definition via Poisson counts),
      superposition and splitting theorems, further properties: times
      between successive events are i.i.d. exponential, conditional
      uniform property.  Heterogeneous processes.

12. Markov processes in continuous time.				XM/SZu
      Definition, transition rates, evolution of probability (forward
      and backward equations).  Jumping chains.  Irreducible
      processes: stationary and limiting distributions, detailed
      balance.  Applications.


STOCHASTIC MODELLING AND SIMULATION

13. Stochastic models I							SF
      Simple population processes, queueing and communication systems, 

14. Stochastic models II						SF
      Random graphs, epidemic models.

15. Simulation techniques.  
      Inverse transform, acceptance based-techniques.			SZa
      Metropolis-Hastings algorithm.  Simulation of stochastic
      processes.


BROWNIAN MOTION AND ELEMENTS OF STOCHASTIC CALCULUS

16. Brownian motion.							IG
      Levy's construction of the Wiener process. Quadratic variation.
      Stochastic integral. Ito's formula. Martingale characterisation
      of the Wiener process.  Martingale representations. Girsanov's
      theorem.

17. Stochastic differential equations (SDEs).				IG
      Notion of solutions.  Existence and uniqueness theorems.
      Stochastic representation of the solutions of elliptic and
      parabolic PDEs. Numerical solutions.

18. Applications of stochastic differential equations.			IG
      Financial models: option pricing. SDEs arising in biology.  A
      brief introduction to filtering and control of stochastic
      systems.

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ASSESSMENT

3-4 homework assignments evenly spaced throughout the course.