Stochastic Processes

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

Venue: LHC 304

Schedule: MON, TUE, & THU 12-13Hr

Lectures

02/01 Discrete-time Markov Chain, Chapman-Kolmogorov equation.
06/01 Definition of Markov processes on a Polish space. An alternative defn of DT MC.
07/01 Continuous-time MC, Embedded chain, Sojourn time, TPM, transition function, Rate matrix, semigroup property of Transition functions. derivation of an Integral equation for transition functions.
09/01 Differentiability of transition function.
13/01 Recapitulation of backward and forward equations. The explosion of a Markov chain. http://web.math.ku.dk/~susanne/kursusstokproc/ContinuousTime.pdf
16/01 Reuter’s criterion
17/01 Recapitulation of some properties of an independent collection of the exponential random variables. Birth & Death process, rate matrix. Application of Reuter's criterion: necessary and sufficient condition of non-explosion.
20/01 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. The Poisson process as a pure birth process. derivation of transition function for a general pure birth process.
21/01 Classification of states, Definition of transient, recurrent and positive recurrent states; irreducibility, aperiodicity.
23/01 Stationary distribution of a Markov process. The convergence of the number of visits per unit time.
27/01 Limit of the empirical measure of an irreducible positive recurrent process.
28/01 An expression of the stationary distribution for an irreducible positive recurrent process.
30/01 Definition of Brownian motion. Finite-dimensional distribution of a stochastic process, co-ordinate process, Kolmogorov's extension theorem.
03/02 Existence of a probability measure on ℝ^[0,∞) such that the coordinate process has independent stationary Gaussian increments.
04/02 The concept of modification, statement of Kolmogorov Centsov Theorem and proof part 1
06/02 Kolmogorov Centsov Theorem proof part 2. Application: construction of Brownian motion as a modification of the coordinate process.
10/02 Quiz 1
11/02 Multidimensional Brownian motion, and its Markov and martingale property.
13/02 Reflection Principle, Distribution of hitting time. Quadratic variation of a deterministic function along a sequence of partitions.
17/02 Revision and tutorial on Brownian motion
18/02 Revision and tutorial on Brownian motion
24/02 Midsem 10-12
02/03 Motivation of stochastic integral, the importance of measuring quadratic variation, optional time and stopping time, Optional sampling theorem (page 6 and Th 3.22 ch1 [3]) [Link]
03/03 Class D and DL, Doob Meyer decomposition, Natural process (4.5, 4.8, 4.10 of ch1 [3])
05/03 quadratic variation of a square-integrable continuous martingale M₂^c. (5.3, 5.8 of ch1 [3]) Equivalent classes of integrands using a measure on the product space.(page 130 of ch 3 [3]).
12/03 A norm on L(M), the space of integrands w.r.t. a given M₂^c integrator. Space of simple processes L0 (Dfn 2.1, 2.3 ch 3 [3])
13/03 (Extra Class) L0, the space of simple processes, stochastic integration I(X) of simple processes. (Dfn 2.3, 2.6 ch 3 [3])
Physical class stops. Online class resumes a month later.
16/04 Recapitulation of the definitions and results.
23/04 The space of square-integrable rcll martingale M₂, and closed subspace M₂^c under || || metric. Integration I is an isometry. (page 137-138 [3])
30/04 Construction of stochastic integral (the L2 Theory), quadratic variation of the integral. (page 138-139 [3])
07/05 Proof of Optional Sampling Theorem (OST.pdf)
14/05 Properties of stochastic integral (page 140-144 [3])
21/05 Stochastic integral w.r.t. continuous local martingales: Definition and properties. Statement of Ito's formula (page 145-147 [3])
28/05 Statement of the Ito's formula and the first part of the proof.
04/06 Remaining part of the proof of Ito's formula.
11/06 Meaning of stochastic differential equations and some examples. (presentation13A.pdf)