 | Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics) by Walter Rudin
Buy used from: $19.50 All the necessary basics of real analysis
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 | Real and Complex Analysis (International Series in Pure and Applied Mathematics) by Walter Rudin
Buy new: $93.75 / Used from: $63.27 More advanced real and complex analysis. Use it for reference mainly.
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 | Introductory Real Analysis by A. N. Kolmogorov
Buy new: $9.50 / Used from: $4.84 A nice classic in real analysis with lots of goodies on linear opperators.
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 | Differential Topology by Victor Guillemin
Buy used from: $99.99 Very nice introduction to differential topology, although quite expensive. The presentation emphasises intuition rather than mathematical formalism and succeeds in instilling a very visual understanding of the material.
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 | Probability and Measure, 3rd Edition by Patrick Billingsley
Buy new: $101.84 / Used from: $85.00 A great classic in measure theoretic probability. Not as easy to read as many other books, but still nice and comprehensive. It sets you up very well for SDE theory, especially the last chapter on Brownian motion and the general theory of stochastic processes.
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 | Probability with Martingales (Cambridge Mathematical Textbooks) by David Williams
Buy new: $44.38 / Used from: $18.89 One day I hope to write a book with the passion, insight, and magnificent pedagogical prowess of this masterpiece. Enough said.
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 | A Course in Probability Theory, Revised Edition, Second Edition by Kai Lai Chung
Buy new: $51.87 / Used from: $68.96 Quite a dinosaur at this point but its a brilliant classic. Well worth struggling through.
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 | Diffusions, Markov Processes, and Martingales: Volume 1, Foundations (Cambridge Mathematical Library) by L. C. G. Rogers
Buy new: $52.30 / Used from: $50.00 Covers advanced Markov chains theory, SDEs, and some of the general theory of stochastic processes. More comprehensive than Karatzas and Shreve but not as well organized in my opinion.
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 | Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics) by Ioannis Karatzas
Buy new: $48.31 / Used from: $57.99 The standard reference on SDE's. Although it is comprehensive, it is not self-contained and one must consult other references for some of the proofs. Read this only if you feel particularly courageous - otherwise, go for Oksendal.
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 | Stochastic Differential Equations: An Introduction with Applications (Universitext) by Bernt Oksendal
Buy new: $40.45 / Used from: $30.00 The standard introductory book to SDE's. Very well written and contains a wealth of applications.
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 | Methods of Mathematical Finance by Ioannis Karatzas
Buy new: $77.95 / Used from: $21.00 The standard reference for mathematical finance. Covers asset pricing, single agent portfolio optimization, general equilibrium, and a little bit of incomplete markets as well.
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 | Introduction to Probability Models, Ninth Edition by Sheldon M. Ross
Buy new: $73.09 / Used from: $52.50 A great book on basic stochastic process with plenty of examples and applications. A nice read and great to come back to every once in a while to brush up on something you may have forgotten.
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 | Markov Chains by Pierre Bremaud
Buy new: $67.45 / Used from: $53.04 Everything you ever wanted to know about Markov Chains. Contains lots of hard-to-find results on convergence rates in addition to being a good overview of the major results in the field.
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 | Dynamic Programming and Optimal Control (2 Vol Set) by Dimitri P. Bertsekas
Buy new: $134.50 / Used from: $196.51 Very nicely written intro to dynamic programming and optimal control. Probably the best there is.
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 | Calculus of Variations by I. M. Gelfand
Buy new: $7.88 / Used from: $6.98 Great introduction to variational calculus in the tradition of old russian applied mathematics: simple, intuitive, and to the point.
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 | A First Course in Optimization Theory by Rangarajan K. Sundaram
Buy new: $36.40 / Used from: $32.98 Everything you ever wanted to know about convex analysis, constrained optimization, Kuhn-Tucker, and even a little bit about dynamic programming. Rigorous but eminently practical.
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 | Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics (Academic Press), 60.) by Morris W. Hirsch
Buy used from: $65.12 A great classic on dynamical systems. Covers everything you need to know about the linear case plus some interesting topics in non-linear dynamics. A bit dated though.
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 | Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in nonlinearity) by Steven H. Strogatz
Buy new: $47.60 / Used from: $32.80 Very nice introduction to the state of the art in dynamics. Very readable.
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 | Lectures on Partial Differential Equations by I. G. Petrovsky
Buy new: $9.95 / Used from: $4.48 Great introduction to partial differential equations. Doesn't do justice to the heat equation, though.
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 | Matrix Analysis by Roger A. Horn
Buy new: $37.80 / Used from: $26.67 This book has a permanent spot on my bookshelf. Although quite old and missing many important topics (e.g. modern applications of SVDs, QZ decomposition...etc.) it contains many important indispensable results. A keeper.
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