Stochastic Processes

Venue: LHC 206

Schedule: MON (10 AM), WED(3PM), THU(10AM)

Lectures

1. 04/01 Motivation. Kolmogorov's model of probability, axiomatic approach. recapitulation of random variable, expectations and conditional expectation. The notion of Markovity.

2. 06/01 Continuous-time pure jump processes, transition probabilities and holding time distributions, semi-Markov processes.Necessary and sufficient condition for that to be Markov.

3. 07/01 Chapman-Kolmogorov equation: semigroup property of Transition functions. For the finite-state Markov process: derivation of an Integral equation for transition functions. The regularity of the Integral equation,

   1. 11/01 Infinitesimal parameters of a Markov process: Q-matrix / rate matrix. Derivation of backward and forward equations for the finite-state case.

4. 13/01 For finite-state: Solution of the backward equation. Rate matrix as a generator of the semigroup of transition functions. Discussion of infinite-state case. Derivation of Integral equation, differentiability and backward equation of transition functions for countable state case.

5. 14/01 Recapitulation of definition of explosion of Markov pure jump processes. Proof of Reuter's criterion for explosion.

6. 18/01 Birth & death processes: rate matrix. Application of Reuter's criterion: necessary and sufficient condition of non-explosion.

7. 20/01 Recapitulation of some properties of independent collection of exponential random variables. Branching process as a B&D process: derivation of forward and backward equation of transition functions.

8. 21/01 Two B&D process: expression of transition function by solving backward equation, study of asymptotic. Recapitulation of definition of Poisson process, Poisson process as a pure birth process. derivation of transition function for a general pure birth process.

9. 23/01 Definition of transient, recurrent and positive recurrent states; irreducibility and stationary distribution of a Markov process. Limit of empirical measure of an irreducible positive recurrent process, expression of stationary distribution.

10. 25/01 Discrete time continuous state Markov and related processes. Time series models.

11. 27/01 Brownian motion, martingale and Markov property of BM. Definition of total variation and quadratic variation (along a sequence of partitions). Results on p-variation of continuous function.

12. 28/01 Levy's Theorem on quadratic variation process of Brownian motion.

13. 01/02 Ito's formula for continuous path with continuous quadratic variation process. (path wise approach by H. Follmer 1979 after fixing a sequence of partitions. Later (on 15/02) we show that the choice is immaterial.)

14. 03/02 Application of Ito's formula to find stochastic integral of functions of Brownian motion, co-variation process, multi-dimensional version of Ito's formula. Definition of filtration, stopping time, martingale and local martingale.

15. 03/02 Definition of Ito integral of an adapted rcll process w.r.t a continuous local martingale. The Ito integral w.r.t a continuous local martingale is a local martingale by assuming Optional Stopping Theorem. (Proof is given on 03/03.)

16. 04/02 Student seminar on Criteria for transience, recurrence and null-recurrence of an irreducible B&D process.

17. 11/02 Quiz 1

18. 15/02 A non-constant continuous local martingale has positive quadratic variation. Quadratic variation is Independent of the choice of the sequence of partitions.

19. 17/02 Ito integral of an adapted rcll process w.r.t a continuous local martingale is Independent of the choice of the sequence of partitions. Recapitulation of conditional probability and conditional expectation.

20. 18/02 Tutorial on assignment 1 and 2, sigma algebra of stopping time.

21. 26/02 Mid semester test.

22. 29/02 Solution key of midsem paper. Approximation of a stopping time by a decreasing sequence of discrete valued stopping times.

23. 02/03 Properties of T-observable σ-field

24. 03/03 Proof of Optional Stopping Theorem, stopped martingale is a martingale.

25. 07/03 Construction of random variable given a cumulative distribution function, construction of independent random variables.

26. 09/03 Finite dimensional distribution of a stochastic process, co-ordinate process, Kolmogorov's extension theorem, existence of Wiener measure.

27. 10/03 Kolmogorov Centsov Theorem, concept of modification.

28. 14/03 Holder continuity of Wiener process.

29. 16/03 L0 the space of simple processes, stochastic integration of simple processes.

30. 17/03 The metric spaces L(M), L*(M) with [] metric. The space of square integrable rcll martingale M₂, and closed subspace M₂^c under || || metric. For M in M₂^c and X in L*(M) definition of I(X), the stochastic integral.

31. 21/03 Student seminar on notions of continuity of stochastic processes.

32. 23/03 Cross variation formula for stochastic integrals.

33. 28/03 Example of an SDE, GBM, solution method and uniqueness.

34. 30/03 Seminar on Feynman Kac formula.

35. 31/03 Theorem: Existence and uniqueness of strong solution to a class of SDEs.

36. 04/04 Use of random measures in the SDEs to obtain rcll solutions.

37. 06/04 Local time of Brownian motion and Tanaka's formula

38. 07/04 Tanaka equation: nonexistence of strong solution, existence and uniqueness of weak solution, Theorem on weakly uniqueness of a class of SDEs.

39. 11/04 Markov and strong Markov property of time homogeneous Ito diffusion.

40. 13/04 Quiz 2

41. 14/04 Infinitesimal generator of time homogeneous Ito diffusion.

42. 22/04 Quiz 3 (Oral)

43. 29/04 End semester test.

Result