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I have taught this course earlier. The web page for one such may be found here.
I have taught this course earlier. The web page for one such may be found here.
Venue:
Schedule: MON & WED at 4PM & THU 11AM
Lectures
08/02 Definition of Markov Process on a Polish space, Different types of Markov Process. An alternative definition of DT MC.
10/02 Continuous-time MC, Sojourn time distribution
11/02 Embedded Markov Chain,
15/02 TPM, transition function, Rate matrix, semigroup property of Transition functions. derivation of an Integral equation for transition functions.
17/02 Differentiability of transition function for countable state MC. Expression of rate matrix, examples.
18/02 The explosion of a Markov chain. Reuter’s criterion http://web.math.ku.dk/~susanne/kursusstokproc/ContinuousTime.pdf
22/02 Birth & Death process, rate matrix. Application of Reuter's criterion: necessary condition of non-explosion.
24/02 Sufficient condition of non-explosion.
25/02 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, the study of asymptotic.
01/03 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.
03/03 Stationary distribution of a Markov process: example, definition and computation. Difference between MC and martingale with example.
04/03 The convergence of the number of visits per unit time. Classification of states, Definition of transient, recurrent and positive recurrent states; irreducibility, aperiodicity. Limit of the empirical measure of an irreducible positive recurrent process. An expression of the stationary distribution for an irreducible positive recurrent process.
08/03 Existence of finite-dimensional random vector given the joint distribution. Finite-dimensional distribution of a stochastic process, co-ordinate process, Kolmogorov's extension theorem.
10/03 Class D and DL, Doob Meyer decomposition, Natural process (4.5, 4.8, 4.10 of ch1 [3]). Quadratic variation of a square-integrable continuous martingale M2c. (5.3, 5.8 of ch1 [3])
11/03 Continuous stopping time and its approximation by a decreasing sequence of discrete stopping times. Process at stopping time is measurable wrt the stopped σ field.
15/03 Optional sampling theorem (OST) for continuous time (page 6 and Th 3.22 ch1 [3])
20/03 Quiz 1
25/03 Midsem (Syllabus: Topics covered till 4th March)
31/03 Stopped martingale is a martingale (Application of OST). 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.
01/04 The concept of modification, statement of Kolmogorov Centsov Theorem (KCT) and proof.
05/04 Tutorial on Quiz and Midsem questions. Application of KCT: Construction of Brownian motion as a modification of the coordinate process in Wiener measure space.
07/04 Quadratic variation of a deterministic function along a sequence of partitions.
08/04 Multidimensional Brownian motion, Reflection Principle, Distribution of hitting time.
12/04 Quiz 2
14/04 Strong Markov property of Brownian motion.
15/04 Equivalent classes of integrands using a measure on the product space.(page 130 of ch 3 [3]). A norm 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]). The space L*(M). Stochastic integration I(X) of simple processes. (Dfn 2.3, 2.6 ch 3 [3]). The || || norm on M2c.
21/04 and 22/04 Construction of stochastic integral of integrands in L*(M) (the L2 Theory). X → I(X) is an isometry.
26/04 Revision in Live session.
28/04 Quadratic variation of the integral. (page 138-139 [3]). Properties of stochastic integral (page 140-144 [3])
29/04 Stochastic integral w.r.t. continuous local martingales: Definition and properties.
Week-long break of all academic activities.
10/05 Revision in Live session and motivation for stochastic differential equation.
12/05 Statement of the Ito's formula and the first part of the proof. (page 145-147 [3])
13/05 Remaining part of the proof of Ito's formula.
17/05 Tutorial on Assignments 4 and 5.
19/05 Stochastic Differential Equation (Definition, Example, and strong solution)
20/05 Stochastic Differential Equation (Weak Solution and Operator associated to that)
24/05 Tutorial
03/06 End-Sem Exam 10:30-12:30 hr
Repeat Exam 15-07-2021 Thursday 03.00 to 05.00
The questions are descriptive in nature. Students are supposed to attach the scanned answers in CT.
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