Stochastic Processes 2023

I have taught this course earlier. The web page for one such can be found here.

Venue: LHC 106

Schedule: MON & THU at 5PM & WED 9AM

Lectures

1. 02/01 Definition of Markov Process on a Polish space, Different types of Markov Process. Continuous-time MC, Time homogeneity. Lec1

2. 04/01 DS-CT Markov Chain. Embedded Markov Chain and the conditional distribution of Sojourn time. Chapman Kolmogorov's equation. Lec2

3. 05/01  TPM, transition function, Rate matrix, semigroup property of Transition functions. Derivation of an Integral equation for transition functions. Differentiability of transition function for countable state MC. Kolmogorov's backward equation. Lec3
   11/01 No class
   12/01 No class

4. 16/01 Continuity of weighted average of uniformly bounded continuous functions on the real line. Expression of rate matrix, examples. Lec4

5. 18/01 The explosion of a Markov chain. Reuter’s criterion  http://web.math.ku.dk/~susanne/kursusstokproc/ContinuousTime.pdf Lec5

6. 19/01 Proof of Reuter’s criterion Lec6

7. 23/01 Birth & Death process, rate matrix. Application of Reuter's criterion: Sufficient condition of non-explosion. Lec7

8.  23/01 Extra Class Necessary condition of non-explosion for B&D process. Recapitulation of some properties of an independent collection of the exponential random variables. Branching process as a B&D process: derivation of the forward and backward equation of transition functions. Two-state B&D processes: expression of transition function by solving the backward equation. Lec8

9. 25/01 The study of asymptotics and stationary distribution of Two-state B&D processes. The Poisson process as a pure birth process. Derivation of transition function for a general pure birth process. Recapitulation of some properties of an independent collection of the exponential random variables. Lec9
   26/01 Holiday

10. 30/01 Stationary distribution of a Markov process: example, definition and computation. Difference between MC and martingale with example. The convergence of the number of visits per unit time. Classification of states, Definition of transient, recurrent and positive recurrent states; irreducibility, aperiodicity. Lec10

11. 01/02 Limit of the empirical measure of an irreducible positive recurrent process. An expression of the stationary distribution for an irreducible positive recurrent process. Computation of stationary distribution of an irreducible B&D process. Lec11

12. 01/02 Extra Class Existence of finite-dimensional random vector given the joint distribution. Finite-dimensional distribution of a stochastic process, co-ordinate process, Kolmogorov's extension theorem. Lec12

13. 02/02 Tutorial and a sketch of proof of Kolmogorov's extension theorem. Lec13
    06/02 Quiz (Max=14.0, Min=2.5, Mean=10.6, Median=12.0, STD=3.9)

14. 08/02 Class D and DL, Doob Meyer decomposition, Natural process, uniqueness of DM decomposition. (4.5, 4.8, 4.10 of ch1 [3]). Lec14

15. 09/02 Quadratic variation of a square-integrable continuous martingale  M2c. (5.3, 5.8 of ch1 [3]) Continuous stopping time and its approximation by a decreasing sequence of discrete stopping times. Lec15

16. 13/02 Process at stopping time is measurable wrt the stopped σ field. Optional sampling theorem (OST) for continuous time (page 6 and Th 3.22 ch1 [3]) Lec16

17. 15/02 Remaining part of the proof of OST.  Lec17

18. 16/02 Stopped martingale is a martingale (Application of OST). Lec18

18/02 10:00 LHC101 Midsem (Max=29.0, Min=17.0, Mean=23.9, Median=24.0, STD=4.2 ) (Syllabus: till Lec16) 

19. 27/02 Definition of Brownian motion and its Markov and martingale properties. Existence of a probability measure on ℝ[0,∞) such that the coordinate process has independent stationary Gaussian increments. Definition of indistinguishability, modification, and Holder continuity Lec19

20. 01/03 Statement of Kolmogorov Centsov Theorem (KCT), and proof. Lec20

21. 02/03 Application of KCT: Construction of Brownian motion as a modification of the coordinate process in Wiener measure space. Lec21

22. 06/03 Quadratic variation of a deterministic function along a sequence of partitions. Quadratic variation of Brownian motion. Lec22
    08/03 Holiday for Holi 

23. 09/03 Multidimensional Brownian motion, Its strong Markov property. Lec23
    13/03 No class as Friday's schedule was followed.

24. 15/03 Reflection Principle, Distribution of hitting time.  Equivalent classes of integrands using a measure on the product space. (page 130 of ch 3 [3]). A metric on L(M), the space of integrands w.r.t. a given M2c integrator. Space of simple processes  L0 (Dfn 2.1, 2.3 ch 3 [3]). Lec24

25. 16/03 The space L*(M). Stochastic integration I(X) of simple processes. (Dfn  2.3,  2.6 ch 3 [3]). The || || metric on M2c.  Lec25

26. 20/03  Tutorial and midsem paper showing.
    22/03 Holiday

27. 23/03 For all X in L0, stochastic integration I(X) is in M2c  and X ↦ I(X)  is an isometry. Lec26

28. 27/03 Construction of stochastic integral of integrands in L*(M) (the L2 Theory). X ↦ I(X)  is an isometry. Lec27. Quadratic variation of the integral, some properties of stochastic integral (page 138-139 [3]). Lec 28

29. 29/03 Quadratic covariation. A nonconstant continuous square integrable martingale has nonzero quadratic variation. Quadratic covariation of two stochastic integrals.(page 140-144 [3]) Lec29

30. 30/03 Stochastic integral w.r.t. continuous local martingales: Definition and properties. (page 145-147 [3]) Lec30

31. 03/04 Statement of Ito's formula and the first part of the proof. (page 148-153 [3]) Lec31

32. 05/04 Remaining part of the proof of Ito's formula. Lec32

33. 06/04 Tutorial: Calculation of quadratic variation of processes driven by Brownian motion by applying Ito's formula. Stochastic Integration by Parts. Lec33

34. 10/04  Geometric Brownian motion and its SDE Lec34

35. 12/04 Stochastic Differential Equation: Definition, Existence and uniqueness of the strong solution. Weak Solution of SDE: Definition and uniqueness. Lec35
    Tanaka's SDE, an example where the strong solution does not exist. Lec36

36. 13/04 Levy's characterization of Brownian motion and the operator associated with the weak solution. Lec37
    14/04 Quiz-2(Max=15.0, Min=6.0, Mean=10.6, Median=11.0, STD=3.7)

37. 17/04 Tutorial
    25/04 End-Sem Exam 10:00-12:00 (Max=38.0, Min=13.0, Mean=29.8, Median=37.0, STD=10.5 )