Analysis:

1. Real Analysis--- H. L. Royde (most accessibleone in real analysis)

2. Measure Theory and Probability Theory---Krishna B.Athreya and Soumendra N. Lahiri

3. A Probability Path ---S.I. Resnick

Stochastic Analysis:

For non-math major:

1.B. Oksendal---Stochastic differential equations (6thEditon)

2.F.C. Klebaner---Introduction to stochasticcalculus with applications (2nd Edition)

3. M.Meyer--Continuous Stochastic Calculus withApplications to Finance

4. Steele M.J.---Stochastic Calculus and FinancialApplications

For Math major :

1.    I.Karatzas, S. E. Shreve---Brownian motion and stochastic calculus (2ndEdition) (Popular in United States)

2.    D.Revuz, M. Yor---Continuous Martingales and Brownian Motion (3rdEdition) (Popular in Europe)

3.    DavidApplebaum---Levy Processes and Stochastic Calculus

Linear Functional Analysis

1.    Sabindranath,Sen---A First Course in Functional Analysis

2.    HaimBrezis---Functional Analysis, Sobolev Spaces and Partial Differential equations

Two Classical books in FunctionalAnalysis:

1. John B. Conway ---A course infunctional analysis (2nd Edition)

2. Peter D. Lax---Functional Analysis

Applications of Real Analysis and FunctionalAnalysis:

1.    AdamBobrowski---Functional Analysis for Probability and Stochastic Processes

2.    AlainBensoussan---Stochastic Control by Functional Analysis Methods

3.    EfeA. OK----Real analysis with economic applications

A Classic book in Partial DifferentialEquations:

Lawrence C. Evans---An Introductionto Stochastic Differential Equations

Stochastic Processes

1.    A first course instochastic processes---Samuel Karlin, Howard M. Taylor

2.    A second course instochastic processes—SamuelKarlin, Howard M. Taylor

3.    Probability and random processes---G. R. Grimmett, D. R. Stirzaker

4.    Probability With Martingales----DavidWilliams

Queueing:

1.    Analysis of Queues:Methods and Applications---N. Gautam (good for fresh beginners)

2.    Convergence of Probability Measures (2nd Edition) ---Patrick Billingsley (This is the systematically theoretic foundation for diffusion approximation in queueing)

3.    Fundamentals of Queueing Networks---Hong Chenand David Yao

(for beginners in queueing network and its approximation)

Optimal Control Theory

Deterministic:

1.   An Introduction to Mathematical Optimal Control Theory-- Lawrence C. Evans(A good note which can be downloaded online)

2.   Optimal Control Theory,Applications to Management Science and Economics (2 nd Edition)---S.P. Sethi, G.L.Thompson

Brownian motion driven:

1.    Stochastic Control inContinuous time---KevinRoss (A short note which can be downloaded online.  It is very good for beginners with knowledgein Ito’s lemma and the idea of DP)

2.    Controlled MarkovProcesses and Viscosity Solutions (2nd Edition) ---Fleming and Soner (Most famous one)

3.    Continuous-Time StochasticControl and Optimization with Financial Applications—Huyen Pham (This is really an excellent book, from which, I learn a lot. )

4.     Brownian Motion and Stochastic flow systems---J.M.Harrison (An excellent small book, which can be downloaded online. It is about instantaneous control of Brownian motion)

There is a 2nd Edition forthe above book and it takes a new title: Brownian Modelsof Performance and Control---JM Harrison(Nov/2013) (This book could not downloaded online, so I do not read it. There is some new things like optimal stopping in this edition)

5.    Applied Stochastic Controlof Jump Diffusions (2nd Edition)---B. Oksendal, A. Sulem

6.    Dynamic Programming andInventory Control--- Alain Bensoussan

Continuous time Markov process driven:

1.    Continuous-Time Markov Decision Processes Theory And Applications---Guo Xianping, Lerma O.H

Point process Driven:

1.    Point processes andqueues, martingale dynamics---Pierre Brémaud

Financial Derivative Pricing:

1.    Options, Futures, andother Derivatives (7th Edition, 2009)---John C. Hull (This book is called as a bible infinancial derivative.  The mathematics here is low-level)

2.    Stochastic Calculus forFinance I & II (2004) --- Shreve, R. E.

(I is about discrete time model, II is about continuous timemodel. This is the most famous book in this area, except Hull’s book.  This book requires the basic knowledge instochastic analysis)

3.    Arbitrage Theory inContinuous Time (3rd Edition) – Tomas Bjork

4.    Dynamic Asset PricingTheory (3rd Edition)—Darrell Duffie

(This is a famous but relatively difficulty book, which is used widely for PHD students in mathematical finance.)

Two good books in Fundamentals of Control Theory:

1.    JerzyZabczyk----Mathematical Control Theory: An Introduction

2.    AndreiAgraghev, Yuri Sachkov---Control Theory from the Geometric Viewpoint

Other excellent books in Finance and Economics:

1.    Principles of FinancialEconomics---S. F. LeRoy, J. Werner.

2.    The Economics of Money,Banking, and Financial markets (7th Edition)---Frederic S. Mishkin (This is an excellent book, little math, but full of logic. After reading most of this book, I can understand the financial news)

3.    Recursive Methods inEconomic Dynamics---NancyL. Stokey, Robert E. Lucas (A lot of Dynamic Programming applications)

4.    Macroeconomics (6th Edition) --- Olivier Blanchard,David R. Johnson