Venue: LHC 106
Schedule: MON & THU at 5PM & WED 9AM
Lectures
02/01 Definition of Markov Process on a Polish space, Different types of Markov Process. Continuous-time MC, Time homogeneity. Lec1
04/01 DS-CT Markov Chain. Embedded Markov Chain and the conditional distribution of Sojourn time. Chapman Kolmogorov's equation. Lec2
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
16/01 Continuity of weighted average of uniformly bounded continuous functions on the real line. Expression of rate matrix, examples. Lec4
18/01 The explosion of a Markov chain. Reuter’s criterion http://web.math.ku.dk/~susanne/kursusstokproc/ContinuousTime.pdf Lec5
19/01 Proof of Reuter’s criterion Lec6
23/01 Birth & Death process, rate matrix. Application of Reuter's criterion: Sufficient condition of non-explosion. Lec7
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
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
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
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
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
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)
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
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
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
15/02 Remaining part of the proof of OST. Lec17
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)
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
01/03 Statement of Kolmogorov Centsov Theorem (KCT), and proof. Lec20
02/03 Application of KCT: Construction of Brownian motion as a modification of the coordinate process in Wiener measure space. Lec21
06/03 Quadratic variation of a deterministic function along a sequence of partitions. Quadratic variation of Brownian motion. Lec22
08/03 Holiday for Holi
09/03 Multidimensional Brownian motion, Its strong Markov property. Lec23
13/03 No class as Friday's schedule was followed.
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
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
20/03 Tutorial and midsem paper showing.
22/03 Holiday
23/03 For all X in L0, stochastic integration I(X) is in M2c and X ↦ I(X) is an isometry. Lec26
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
29/03 Quadratic covariation. A nonconstant continuous square integrable martingale has nonzero quadratic variation. Quadratic covariation of two stochastic integrals.(page 140-144 [3]) Lec29
30/03 Stochastic integral w.r.t. continuous local martingales: Definition and properties. (page 145-147 [3]) Lec30
03/04 Statement of Ito's formula and the first part of the proof. (page 148-153 [3]) Lec31
05/04 Remaining part of the proof of Ito's formula. Lec32
06/04 Tutorial: Calculation of quadratic variation of processes driven by Brownian motion by applying Ito's formula. Stochastic Integration by Parts. Lec33
10/04 Geometric Brownian motion and its SDE Lec34
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
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)
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 )