Linear Algebra

1. Vector spaces, bases, linear transformations, matrices.
2. 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).
3. Iterative methods for solving linear equations---Gauss-Seidel, successive over-relaxation, …
4. Orthogonal matrices, discrete least square problem, Gram-Schmidt method.
5. 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).
6. Introduction to optimization---gradient methods.

Suggested Textbook

1. 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

1. W. Rudin: Principles of mathematical analysis, Tata McGraw-Hill.

Probability and Statistics I

Prerequisite: Real Analysis

1. Combinatorial probability, Independence of events, Conditional probabilities, - Hands on with R
2. Random variables, densities, Expectation, Variance and moments, Standard univariate distributions, Independence of random variables, Moment Generating Functions
3. Tchebychev's inequality and weak law of large numbers, Central Limit Theorem. - Hands on with R
4. Marginal Distribution, Conditional Distribution, Conditional expectation, Regression, Correlation, Bivariate normal distribution, Multivariate normal distribution, Copula Models - Hands on with R
5. 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

1. R. Ash: Basic Probability Theory, : John Wiley & Sons (1970).
2. P. Billingsley: Probabilty and Measure, Third Edition, John Wiley & Sons (1995).
3. W. Feller: Introduction to Probability Theory and its Applications, Volume 1, Third Edition, John Wiley & Sons (1972).
4. P.G. Hoel, S.C. Port & C.J. Stone: Introduction to Probability Theory Houghton-Miffin (1971).
5. G.K. Bhattacharya & R.A. Johnson: Statistics : Principles and Methods, Second Edition, John Wiley & Sons (1992).
6. P.J. Bickel & K.A. Doksum: Mathematical Statistics, Holden-Day, (1977).
7. P. G. Hoel, S. C. Port, and C. J. Stone: Introduction to Statistical Theory, Houghton Mifflin (1971).
8. George Casella and Roger L. Berger: Statistical Inference (Second Ed.), S Chand & Co (2001).

Probability and Statistics II

Prerequisite: Probability and Statistics I

1. Sampling distributions based on normal populations - t, chi-square and F distributions - Hands on with R
2. 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
3. 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
4. Order statistics, empirical distribution function, Glivenko-Cantelli Theorem (statement only). - Hands on with R
5. 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
6. 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

1. Regression model, ANOVA models, linear models, ordinary least square, Sampling distribution of regression estimates, Gauss-Markov theorem, testing linear restrictions. - Hands on with R
2. Autocorrelations and heteroskedasticity, Instrumental variables and simultaneous equation models. Structural and reduced form models. - Hands on with R
3. 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

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

1. P. Billingsley : Probabiltiy and Measure, Third Edition, John Wiley & Sons (1995).

Discrete mathematics

1. Some counting principles
2. Basic logic
3. Finite automata and regular languages

Suggested Textbooks

1. N.L. Biggs : Discrete Mathematics, Oxford Science Publications.
2. J. Nesetril, J. Matousek: Invitation to Discrete Mathematics, Clarendon Press.
3. M. Huth and M. Ryan: Logic in computer science, Cambridge University Press (2005).
4. 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

1. G.F. Simmons: Differential Equations With Applications and Historical Notes 2e, Tata McGraw-Hill.

Algorithms

Suggested Textbooks

1. T.H. Cormen, C.E. Leiserson, and R.L. Rivest: Introduction to algorithms, Prentice-Hall (1998).
2. J. Kleinberg and E. Tardos: Algorithm design, Pearson/Addison-Welsey (2006).
3. C. Papadimitriou and K. Steiglitz: Combinatorial Optimization

Economics

1. Theory of consumer choice. Utility theory and preferences. Demand, revealed preferences, comparative statics.
2. Extension of basic choice models to include time and uncertainty.
3. Markets. Perfect competition, Monopoly, Monopolistic competition (Dixit-Stiglitz). Walrasian equilibrium.
4. Risk aversion, risk sharing. Contingent claims.
5. Game theory. Introduction to cooperative and non-cooperative games.
6. Introduction to externalities and market failures.
7. Adverse selection, moral hazard, principal-agent contracts.
8. Introduction to auctions.
9. Macroeconomics: Aggregate consumption, aggregate investment, money and financial markets, introduction to components of national income accounts.
10. IS-LM in a closed economy.
11. IS-LM analysis in an open economy

Suggested Textbooks

1. Hal Varian: Intermediate Microeconomics: A Modern Approach, 7th ed.
2. Olivier Blanchard: Macroeconomics, 5th ed.
3. 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

Econometrics

Prerequisite: Probability and Statistics I and II

1. 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).
2. 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.
3. Value at Risk (VaR): Econometric approach to VaR calculation, RiskMetrics (of J P Morgan).
4. 4. Vector Autoregressive (VAR) models, Cointegration and forecasting for cointegrated VAR models.

Regression and Classification

Prerequisite: Probability and Statistics I and II

1. Least-Square Method, Overview of Supervised and Unsupervised Learning, Concepts of Model fitting, Model Validation and Model Testing, Training, Validation and Test Data
2. 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
3. Classification, Linear Classifiers, Linear Discriminant Analysis (LDA), Logistic Regression, Naive Bayes Classifier, Decision Tree, CART, CHAID, Bagging, Boosting and Random Forest
4. Cluster Analysis, K-means Clustering, Hierarchical Clustering, Model Based Clustering

Finance

Prerequisite: Real Analysis, Probability and Statistics I

1. Financial Statement Analysis :Balance Sheet and Income statement, Cash Flow Statement, Corporate Taxes Financial Ratio Analysis, Valuation of financial assets.
2. Time value of Money, Introduction to Primary Securites, Bonds and Equity, Risk free rate of interest, Financial Returns, Net Return, Log Return, Compounding, Annuities
3. 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
4. Portfolio Theory, Efficient Frontier, CAPM, Asset pricing models
5. Hands on practical with R

Suggested Textbooks

1. Steven E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer
2. Petters, A. O., and Dong, X., An Introduction to Mathematical Finance with Applications, Springer
3. Ross, Sheldon, An Elementary Introduction to Mathematical Finance, Second Edition, Cambridge University Press

Mathematical Finance

Prerequisite: Probability and Statistics I,II and Finance

1. 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
2. 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
3. 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
4. Hands on practical with R

Suggested Textbooks

1. Steven E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer
2. Steven E. Shreve, Stochastic Calculus for Finance II: Continuous Time Models, Springer
3. Chapter 6 of Kallianpur and Karandikar, Introduction to Option Pricing Theory
4. Petters, A. O., and Dong, X., An Introduction to Mathematical Finance with Applications, Springer
5. Ross, Sheldon, An Elementary Introduction to Mathematical Finance, Second Edition, Cambridge University Press

Financial risk management

Prerequisite: Finance and Mathematical Finance

Market risk

1. Value-at-Risk, Expected Shortfall, Max Drawdown. Estimating VaR including extreme value theory, copula model and simulation method - Hands on with R
2. Pricing Asian options and Exotic options using Monte Carlo methods - Hands on with R
3. Estimation of Yield-curve and its effect in Bond pricing, Risk in Bond price via interest rate volatility - Hand on with R

Credit risk

1. Credit ratings, events of default, default probabilities.
2. Structural and Reduced Form models of credit risk.
3. Structural models: Merton and KMV models. Bond models: credit spreads
4. Logistic regression and machine learning models for predicting default - Hands on with R
5. Credit derivatives and its limitations

Suggested Textbooks

1. Bodi Zvi, Alex Kane, Alan J Marcus and Pitabas Mohanty, Investments, McGraw Hill 8th Ed.
2. Richard A. Brealey , Stewart C. Myers Franklin Allen and Pitabas Mohanty, Principles of Corporate Finance, McGraw Hill 8th Edition.
3. William Sharpe , Gordon J. Alexander and Jeffery V. Bailey, Investments, Prentice Hall 5th Ed.
4. Edward I. Altman And Gabriele Sabato: Modelling Credit Risk for SMEs: Evidence from the U.S. Market
5. John C. Hull: Options, Futures and Other Derivatives. Pearson Publications.
6. Frank Fabozzi: Handbook of Fixed Income Securities.
8. Thomas E. Copeland and J. Fred Weston: Financial Theory and Corporate Policy , Addison-Wesley Publishing Company 3rd Ed.

Applied Statistics

Advanced Algorithms

Suggested Textbooks

1. M. Garey and D. Johnson: Computers and Intractability -- the theory of NP-completeness.
2. R. Motwani and P. Raghavan: Randomized Algorithms, Cambridge University Press (1995).
3. V. Vazirani: Approximation Algorithms, Springer (2001).
4. Rolf Niedermier: Invitation to fixed parameter algorithms, Oxford University Press (2006).

Algorithms on Strings, trees and sequences

Suggested Textbook

1. Dan Gusfeld: Algorithms on Strings, Trees and Sequences, Cambridge University Press (1997).

Data Mining and Machine Learning

Suggested Textbooks

1. Jiawei Han, Micheline Kamber: Data mining: concepts and techniques (2nd ed), Morgan Kaufman (2006).
2. Bing Liu: Web Data Mining: Exploring Hyperlinks, Contents and Usage Data, Springer (2006).
3. Soumen Chakrabarti: Mining the Web: Discovering knowledge from hypertext data, Elsevier (2003).
4. Christopher D Manning, Prabhakar Raghavan and Hinrich Schütze : An Introduction to Information Retrieval, Cambridge University Press (2009).

Cryptography

Suggested Textbooks

1. N. Koblitz: A course in number theory and cryptography, GTM, Springer.
2. S. C. Coutinho: The Mathematics of Ciphers, A. K. Peters.
3. D. Welch: Codes and Cryptography.
4. W. Stallings: Cryptography and Network Security.

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

1. J. E. Hopcroft and J. D. Ullman: Introduction to Automata theory, Languages and Computation, Narosa.
2. 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

1. John C Mitchell, Concepts in Programming Languages, Cambridge University Press, 2003.
2. R. Sethi: Programming Languages Concepts and Constructs, Addison-Wesley.