-
Linear Algebra
- Vector spaces, bases, linear transformations,
matrices.
- Direct methods for solving linear equations---Gaussian
elimination, positive definite systems, LU decomposition
(by row-column operations), Banded systems (matrices
whose nonzero entries are in a narrow band).
- Iterative methods for solving linear
equations---Gauss-Seidel, successive over-relaxation,
…
- Orthogonal matrices, discrete least square problem,
Gram-Schmidt method.
- Eigenvalues, reduction to standard forms, the QR
algorithm, its implementation, generalized eigenvalue
problem, Jordan canonical form, something on non-square
matrices like decomposition of matrices by orthogonal
transformations on domain and range (singular value
decompositions).
- Introduction to optimization---gradient methods.
Suggested Textbook
- P.G. Ciarlet: Introduction To Numerical Linear Algebra
And Optimisation, Cambridge University Press.
-
Analysis
Finite, countable and uncountable sets---metric spaces---compact
sets---perfect sets---connected sets---convergent and Cauchy
sequences ---power series---absolute
convergence---rearrangements---limits--- continuity and
compactness---connectedness---discontinuities---differentiablity---Riemann
integration ---sequences and series of functions---equicontinuous
family of functions and the Stone-Weierstrass theorem.
Suggested Textbook
- W. Rudin: Principles of mathematical
analysis, Tata McGraw-Hill.
-
Probability and Statistics I
Prerequisite: Real Analysis
- Combinatorial probability, Independence of events,
Conditional probabilities, - Hands on with R
- Random variables, densities, Expectation, Variance and
moments, Standard univariate distributions, Independence of
random variables, Moment Generating Functions
- Tchebychev's inequality and weak law of large numbers,
Central Limit Theorem. - Hands on with R
- Marginal Distribution, Conditional Distribution,
Conditional expectation, Regression, Correlation, Bivariate
normal distribution, Multivariate normal distribution, Copula
Models - Hands on with R
- Introduction to Statistics with examples of its use, Draw
random samples, Descriptive statistics, Graphical statistics:
Histogram, scatter diagram, Pie diagram, estimates sample
moments, sample mean, sample standard deviation, Hands-on with
R
Suggested Textbooks
- R. Ash: Basic Probability Theory, : John Wiley & Sons
(1970).
- P. Billingsley: Probabilty and Measure, Third Edition,
John Wiley & Sons (1995).
- W. Feller: Introduction to Probability Theory and its
Applications, Volume 1, Third Edition, John Wiley & Sons (1972).
- P.G. Hoel, S.C. Port & C.J. Stone: Introduction to
Probability Theory Houghton-Miffin (1971).
- G.K. Bhattacharya & R.A. Johnson: Statistics : Principles
and Methods, Second Edition, John Wiley & Sons (1992).
- P.J. Bickel & K.A. Doksum: Mathematical Statistics,
Holden-Day, (1977).
- P. G. Hoel, S. C. Port, and C. J. Stone: Introduction to
Statistical Theory, Houghton Mifflin (1971).
- George Casella and Roger L. Berger: Statistical
Inference (Second Ed.), S Chand & Co (2001).
-
Probability and Statistics II
Prerequisite: Probability and Statistics I
- Sampling distributions based on normal populations - t,
chi-square and F distributions - Hands on with R
- Sufficient and minimal sufficient statistics. Point and
Interval Estimation, Consistency, Minimum Variance Unbiased
Estimator (statement only). Theory and Methods of Estimation,
method of moments estimators, maximum likelihood estimator,
consistency and asymptotic normality of MLE's (statement
only) - Hands on with R
- Testing of Hypothesis: one sample and two sample tests
based on t, chi-square and F distributions. - Error
probabilities, statistical power of test, p-values,
log-likelihood ratio test - Hands on with R
- Order statistics, empirical distribution function,
Glivenko-Cantelli Theorem (statement only). - Hands on with
R
- Nonparametric confidence intervals for Quantiles and
confidence bands the distribution function, Chi-square and
Kolmogorov Goodness-of-fit Tests, Sign and Signed Rank Test,
Wilcoxon-Mann-Whitney tests, Kruskal-Wallis Test, Bootstrap
and Resample Techniques - Hands on with R
- Bayesian Methods, Prior distribution, Posterior
Distribution, Conjugate Prior for Binomial, Poisson and
Normal Distribution, Introduction to Hierarchical Bayesian
Models (only normal models) - Hands on with R
-
Probability and Statistics III
Prerequisite: Probability and Statistics I and II
- Regression model, ANOVA models, linear models, ordinary
least square, Sampling distribution of regression estimates,
Gauss-Markov theorem, testing linear restrictions. - Hands on
with R
- Autocorrelations and heteroskedasticity, Instrumental
variables and simultaneous equation models. Structural and
reduced form models. - Hands on with R
- Time series models. Trend, Seasonality,
Forecasting. Linear trend, log-linear trend, Autoregressive
Model, AR(p), Maximum likelihood estimation, Generalized
Method of Moments. - Hands on with R
-
Programming Techniques
Introduction to basic programming principles using Python, including
object-oriented design, big-oh notation, sorting and search
algorithms, elementary data structures (lists, heaps, binary trees).
Suggested Textbooks
- Mark Pilgrim : Dive into Python,
available online.
- T.H. Cormen, C.E. Leiserson, and R.L. Rivest : Introduction
to algorithms, Prentice-Hall (1998).
-
Measure Theoretic Probability
Measure and integration: sigma fields and monotone class theorem,
probability measures, statement of Caratheodory extension theorem,
measurable functions, integration, Fatou, MCT, DCT, product spaces,
Fubini. (about 1/2 time to be spent) Probability: 1-1 correspondence
between distribution functions and probabilities on R, independence,
Borel-Cantelli, Weak and Strong laws in the i.i.d. case, Kolmogorov
0-1 law, various modes of convergence, characterstic functions,
uniqueness/inversion/Levy continuity theorems, CLT for the iid case
with finite variance. (about 1/2 time to be spent)
Suggested Textbook
- P. Billingsley : Probabiltiy and Measure, Third Edition,
John Wiley & Sons (1995).
-
Discrete mathematics
- Some counting principles
- Basic logic
- Finite automata and regular languages
Suggested Textbooks
- N.L. Biggs : Discrete Mathematics, Oxford
Science Publications.
- J. Nesetril, J. Matousek: Invitation to Discrete
Mathematics, Clarendon Press.
- M. Huth and M. Ryan: Logic in computer
science, Cambridge University Press (2005).
- D. Kozen : Automata and Computability, Springer.
-
Differential Equations
Solution of First-order ODE's, Linear ODE's, Especially Second Order with
Constant Coefficients; Undetermined Coefficients and Variation of
Parameters; Sinusoidal and Exponential Signals: Oscillations, Damping,
Resonance; Fourier Series, Periodic Solutions; Delta Functions, Convolution,
and Laplace Transform Methods; Matrix and First-order Linear Systems:
Eigenvalues and Eigenvectors;
Suggested Textbook
- G.F. Simmons: Differential Equations With Applications and
Historical Notes 2e, Tata McGraw-Hill.
-
Algorithms
A quick revision of sorting, searching, selection and Big Oh;
Divide and Conquer;
Dynamic Programming;
Graphs, BFS, DFS, connectivity;
Algorithms on Matrices;
Combinatorial Optimization --- Linear Programming,
Simplex, Duality,
Primal Dual Algorithms
(shortest paths, max flow, matching).
Suggested Textbooks
- T.H. Cormen, C.E. Leiserson, and R.L. Rivest: Introduction
to algorithms, Prentice-Hall (1998).
- J. Kleinberg and E. Tardos: Algorithm design,
Pearson/Addison-Welsey (2006).
- C. Papadimitriou and K. Steiglitz: Combinatorial
Optimization
-
Economics
This course covers material useful for an understanding of both
theoretical and empirical finance. It is not intended as a
comprehensive survey of economics. The approach is analytical (as
befits a graduate math course) and stresses understanding of concepts.
Topics from both micro and macro economics are covered.
- Theory of consumer choice. Utility theory and preferences.
Demand, revealed preferences, comparative statics.
- Extension of basic choice models to include time and
uncertainty.
- Markets. Perfect competition, Monopoly, Monopolistic
competition (Dixit-Stiglitz). Walrasian equilibrium.
- Risk aversion, risk sharing. Contingent claims.
- Game theory. Introduction to cooperative and non-cooperative
games.
- Introduction to externalities and market failures.
- Adverse selection, moral hazard, principal-agent contracts.
- Introduction to auctions.
- Macroeconomics: Aggregate consumption, aggregate investment,
money and financial markets, introduction to components of national
income accounts.
- IS-LM in a closed economy.
- IS-LM analysis in an open economy
Suggested Textbooks
- Hal Varian: Intermediate Microeconomics: A Modern Approach, 7th
ed.
- Olivier Blanchard: Macroeconomics, 5th ed.
- Rudiger Dornbusch, Stanley Fischer, and Richard Startz:
Macroeconomics, 9th ed.
-
Stochastic Processes I
- Markov chains (discrete time, discrete space).
- Conditional expectation. Discrete parameter martingales with
applications,
- Introduction to continuous parameter stochastic processes,
Brownian motion and Poisson process- definition and elementary
properties.
- Stochastic integral wrt Brownian motion and Ito formula.
-
Computational methods
- Numerical analysis: numerical integration and numerical
differentiation.
- Numerical solutions to differential equations.
- Introduction to software for numerical computaion (Octave or Gnu
Scientific Library)
-
Simulation techiques
Pseudo random numbers, Linear congruential generator, Mersenne twister
RNG, Simulation of random variables, illustrations, Monte Carlo
integration, Simulation of Random walk and approximations to diffusion
processes, Applications to credit risk, complex derivatives pricing
and portfolio optimization.
-
Econometrics
Prerequisite: Probability and Statistics I and II
- Non-linear Time Series: Threshold Autoregressive (TAR)
model, Smooth Transition Autoregressive (STAR) model,
Conditional Heteroskedasticity models for volatility (ARCH,
GARCH, and their variants), Regime Switching models for
return and volatility (based on
observables/unobservables).
- High Frequency Data (Transaction Level Data) Analysis: Empirical characteristics of transactions data, Models for non-synchronous trading, Bid-Ask Spread, Price Changes. Bivariate models for price change and duration.
- Value at Risk (VaR): Econometric approach to VaR calculation, RiskMetrics (of J P Morgan).
- 4. Vector Autoregressive (VAR) models, Cointegration and forecasting for cointegrated VAR models.
-
Regression and Classification
Prerequisite: Probability and Statistics I and II
- Least-Square Method, Overview of Supervised and
Unsupervised Learning, Concepts of Model fitting, Model
Validation and Model Testing, Training, Validation and Test
Data
- Regression, Linear Models, Gauss-Markov Theorem,
Multiple Regression, Variable Selection, Bayesian Linear
Regression, Ridge Regression, LASSO, Elastic Net, Principal
Component Regression, Functional Regresiion, Spline
Regression
- Classification, Linear Classifiers, Linear
Discriminant Analysis (LDA), Logistic Regression, Naive Bayes
Classifier, Decision Tree, CART, CHAID, Bagging, Boosting and
Random Forest
- Cluster Analysis, K-means Clustering, Hierarchical
Clustering, Model Based Clustering
Finance
Prerequisite: Real Analysis, Probability and Statistics I
- Financial Statement Analysis :Balance Sheet and Income
statement, Cash Flow Statement, Corporate Taxes Financial
Ratio Analysis, Valuation of financial assets.
- Time value of Money, Introduction to Primary Securites,
Bonds and Equity, Risk free rate of interest, Financial
Returns, Net Return, Log Return, Compounding, Annuities
- Discounting, Zero Coupon Bond and Regular Bond,
Fundamentals of Bond Valuation, Spot Rate Curve, Yield Curve,
Clean and Dirty Price of Bond, Term Structure, Pricing Yield
Curve with Nelson-Siegel Model, Simulate Bond Prices
- Portfolio Theory, Efficient Frontier, CAPM, Asset pricing
models
- Hands on practical with R
Suggested Textbooks
- Steven E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer
- Petters, A. O., and Dong, X., An Introduction to Mathematical Finance with Applications, Springer
- Ross, Sheldon, An Elementary Introduction to Mathematical Finance, Second Edition, Cambridge University Press
Mathematical Finance
Prerequisite: Probability and Statistics I,II and Finance
- Conditional Expectation, Martingales, Markov Processes,
Change of Measure, Radon-Nikodym Derivative, Random Walk,
First Passage Times, Reflection Principle, Brief Introduction
to Stocastic Calculus upto Ito-formula
- Binomial Asset Pricing Model (one and multiperiod model),
No arbitrage, Q-Martingale, Fundamental Theorem of Asset
Pricing, Geometric Brownian Motion as limit of Binomial Asset
Pricing Model
- Introduction to Derivatives, Futures and Option, European
and American Options, Risk-Neutral Pricing, Martingale
Represeantation Theorem, Black-Scholes formula for European
Options, Non-path-dependent American Derivatives, Stopping
Times, General Amarican Derivatives, American Call Options,
Evaluating derivatives via Binomial Option Pricing
Models
- Hands on practical with R
Suggested Textbooks
- Steven E. Shreve, Stochastic Calculus for Finance I:
The Binomial Asset Pricing Model, Springer
- Steven E. Shreve, Stochastic Calculus for Finance II:
Continuous Time Models, Springer
- Chapter 6 of Kallianpur and Karandikar, Introduction
to Option Pricing Theory
- Petters, A. O., and Dong, X., An Introduction to
Mathematical Finance with Applications, Springer
- Ross, Sheldon, An Elementary Introduction to Mathematical
Finance, Second Edition, Cambridge University Press
-
Financial risk management
Prerequisite: Finance and Mathematical Finance
Market risk
- Value-at-Risk, Expected Shortfall, Max
Drawdown. Estimating VaR including extreme value theory,
copula model and simulation method - Hands on with R
- Pricing Asian options and Exotic options using Monte
Carlo methods - Hands on with R
- Estimation of Yield-curve and its effect in Bond pricing,
Risk in Bond price via interest rate volatility - Hand on with R
Credit risk
- Credit ratings, events of default, default
probabilities.
- Structural and Reduced Form models of credit risk.
- Structural models: Merton and KMV models. Bond models:
credit spreads
- Logistic regression and machine learning models for
predicting default - Hands on with R
- Credit derivatives and its limitations
Suggested Textbooks
- Bodi Zvi, Alex Kane, Alan J Marcus and Pitabas Mohanty,
Investments, McGraw Hill 8th Ed.
- Richard A. Brealey , Stewart C. Myers Franklin Allen and
Pitabas Mohanty, Principles of Corporate Finance,
McGraw Hill 8th Edition.
- William Sharpe , Gordon J. Alexander and Jeffery
V. Bailey, Investments, Prentice Hall 5th Ed.
- Edward I. Altman And Gabriele Sabato: Modelling
Credit Risk for SMEs: Evidence from the U.S. Market
- John C. Hull: Options, Futures and Other
Derivatives. Pearson Publications.
- Frank Fabozzi: Handbook of Fixed Income
Securities.
-
- Thomas E. Copeland and J. Fred Weston: Financial
Theory and Corporate Policy , Addison-Wesley Publishing
Company 3rd Ed.
-
Applied Statistics
Correlations, significance, regression (logistic, linear, time
series), testing of hypotheses, clustering techniques, factor
analysis, principal component analysis, distributions (parametric and
non parametric).
-
Advanced Algorithms
This course is about techniques for dealing with algorithmically hard
problems.
NP completeness; Approximation Algorithms (including LP rounding and
primal dual algorithms); Randomized algorithms; fixed parameter
algorithms; branch and bound, local search; heuristics
Suggested Textbooks
- M. Garey and D. Johnson: Computers and Intractability -- the theory
of NP-completeness.
- R. Motwani and P. Raghavan: Randomized Algorithms,
Cambridge University Press (1995).
- V. Vazirani: Approximation Algorithms, Springer (2001).
- Rolf Niedermier: Invitation to fixed parameter
algorithms, Oxford University Press (2006).
-
Algorithms on Strings, trees and sequences
This is a course on topics in algorithms oriented towards applications
in Biology.
String Matching and variations;
Suffix trees and its uses;
Inexact matching;
Sequence Alignment;
Sequence databases and their uses
Suggested Textbook
- Dan Gusfeld: Algorithms on Strings, Trees and Sequences,
Cambridge University Press (1997).
-
Data Mining and Machine Learning
Association rules, frequent itemsets;
Finding high-correlation with low-support;
Classifiers -- Bayesian, Nearest Neighbour;
Decision Trees;
Clustering techniques;
Vector space (TF-IDF) model;
Stop words and stemming;
Supervised learning : Bayesian Networks, Support Vector Machines;
Semisupervised learning: Expectation maximization;
Web search: HITS and PageRank;
Suggested Textbooks
- Jiawei Han, Micheline Kamber: Data mining: concepts and
techniques (2nd ed), Morgan Kaufman (2006).
- Bing Liu: Web Data Mining: Exploring Hyperlinks,
Contents and Usage Data, Springer (2006).
- Soumen Chakrabarti: Mining the Web: Discovering knowledge
from hypertext data, Elsevier (2003).
- Christopher D Manning, Prabhakar Raghavan and
Hinrich Schütze : An Introduction to
Information Retrieval, Cambridge University Press
(2009).
-
Cryptography
Elementary number theory --- Pseudo-random bit generation ---
elementary cryptosystems --- number theoretic algorithms (RSA) ---
symmetric key cryptosystems - DESIDEA, AES, --- authentication ---
digital signatures, electronic commerce (anonymous cash,
micropayments), key management--- PGP --- zero-knowledge protocols ---
fairness.
Suggested Textbooks
- N. Koblitz: A course in number theory and cryptography, GTM,
Springer.
- S. C. Coutinho: The Mathematics of Ciphers, A. K. Peters.
- D. Welch: Codes and Cryptography.
- W. Stallings: Cryptography and Network Security.
- Theory of Computation
Finite automata---regular languages---pumping lemma---
stack automata---context free languages---applications to
compilers---Turing machines---universal Turing machines---
halting problem---non deterministic Turing machines---
complexity classes---P v/s NP.
Recommended Texts
- J. E. Hopcroft and J. D. Ullman: Introduction to Automata theory,
Languages and Computation, Narosa.
- D. Kozen: Automata and Computability, Springer.
-
Programming Language Concepts
Imperative programming---Scope rules---Object
oriented-programming--- Java---Shell
programming---PERL--- Functional programming---Logic
programming---Query Language for databases.
Laboratory: Programming assignments in Java, PERL and SQL.
Recommended Texts
- John C Mitchell, Concepts in Programming Languages, Cambridge University Press, 2003.
- R. Sethi: Programming Languages Concepts and Constructs, Addison-Wesley.