Statistics ( Ph.D. )

1. Training/Research Orientation

• Markov Process and Stochastic Differential Equations
• Stochastic Control and Financial Risk
• Mathematical Statistics and Applications
• Quantitative Economic Analysis
• Statistics Learning and Applications

2. Program Duration and Credit

Three years in general, the maximum duration should not exceed 7 years (including the extension time).
20 credits in total, at least 12 compulsory credits.

3. Core Courses and Introduction

Stochastic Analysis
Stochastic Analysis mainly talks about all kinds of definitions of conditional expectation, discrete-time martingale, continuous-time martingale and stochastic integral, and theories of continuous semi-martingale, exponential martingale, Ito formula and Girsanov theorem. Through the study of this course, students can use martingale theories or methods to analyze and deal with complex stochastic processes or models, and further deepen understanding and application of martingale and stochastic integral.
The course mainly discusses theories of modern martingale and stochastic integral, including discrete-time martingale, continuous-time martingale, uniformly integrable martingale, continuity of martingale, martingale convergence theorem, inequalities of martingale, martingale decomposition theorem, Doob stopping theorem, predictable stochastic process, Doleans measure, stochastic integral of predictable process with respect to  martingale, stochastic integral of adapted process, square integrable martingale, quadratic variation process on continuous local martingale, Ito formula on continuous semi-martingale, Girsanov theorem, equivalent martingale measure, weak predictable representation of the Brownian motion, the Tanaka formula, strong solution and weak solution of SDE, the martingale problem of SDE, L diffusion process and Markov property, square integrable martingale decomposition, square integrable martingale representation theorem, conditional expectation martingale representation theorem, Fubini theorem, functional structure of diffusion process, the Black-Scholes formula and so on.

Stochastic Differential Equations
Stochastic Differential Equations mainly introduces concepts of Ito integral, Ito formula, stochastic differential equation (SDE), Ito diffusion process and Dynkin formula, and corresponding theories and applications. By studying basic theories of stochastic differential equations, doctor graduate students are normally expected to obtain the ability of scientific thinking, scientific innovation, as well as to solve practical problems.
The course mainly shows basic theories of SDEs and corresponding methods, including probability space, conditional expectation, uniformly integrability, martingale convergence theorem, random variables, stochastic process, Brownian motion, construction of the Ito integral, extensions of the Ito integral, (one-dimensional or multi-dimensional) Ito formula, the martingale representation theorem, existence and uniqueness of the solution of SDEs, weak and strong solutions, Markov property, strong Markov property, the generator of an Ito diffusion, the Dynkin formula, the characteristic operator, Kolmogorov’s backward equation, the Feynman-Kac formula, the martingale problem, Girsanov theorem, the Hamilton-Jacobi-Bellman equation, (one-dimensional or multi-dimensional) linear filtering problem, the Dirichlet-Poisson problem, boundary value problem, optimal stopping problem, financial market, optimal portfolio.

Qualitative Theory of Stochastic Differential Equations
Qualitative Theory of Stochastic Differential Equations introduces theories and methods of stochastic differential equations, including existence and uniqueness of stochastic differential equations, autonomous stochastic differential systems, non-autonomous stochastic differential systems and dynamical systems. Doctor graduate students are expected to grasp basic theories for stability of stochastic system, master methods to stabilize stochastic systems and apply these methods to deal with real models after learning this course.
The course mainly discusses related literature on stochastic stability. The content of this course includes:

• basic theories: concepts of stability of stochastic systems and corresponding properties, stability of stochastic linear systems, Lyapunov stability theorem, stability of stochastic nonlinear systems, stability and regularity of stochastic dynamical systems, pth moment stability of stochastic systems, pth moment asymptotic stability, pth moment exponential stability
• finite-time stability: concept of finite-time stability and its properties, finite-time stability theorem and its extension, finite-time stability of autonomous systems, finite-time stability in probability of stochastic nonlinear systems, back-stepping design method and its applications in financial models
• global asymptotic stability: concept of global asymptotic stability and its properties, global asymptotic stability theorem, global stability of stochastic systems with time-varying delays, relations between finite-time stability and global asymptotic stability, applications of global asymptotic stability in financial models
• stability of regime-switching stochastic systems: stability of regime-switching diffusion, optimal of regime-switching risk process
• approximation of the solution of SDEs: the fundamental mean-square convergence theorem for SDEs

Stochastic Control Theory
Stochastic Control Theorymainly introduces stochastic differential equation with control variables, dynamic programming approach, Hamilton-Jacobi-Bellman equation, classical solution or viscosity solution of the HJB equation, and verification theorem. Doctor graduate students are expected to grasp the basic concepts of stochastic control theory as well the applications in insurance financial control. After learning this course, their abilities of scientific thinking and scientific innovation as well as to solve practical problems must be fostered and enhanced.
The course mainly discusses basic theories and methods of stochastic control theory, including:

• Stochastic calculus with jump diffusions
• Optimal stopping of jump diffusions
• Stochastic control of jump diffusions
• Combined optimal stopping and stochastic control
• Singular control for jump diffusions
• Impulse control of jump diffusions
• Combined stochastic control and impulse control
• Viscosity solution

Stochastic Partial Differential Equation
Stochastic Partial Differential Equation, which emerged in 1960s, is one of professional courses on Markov process and stochastic differential equations. This course mainly introduces the solution properties of stochastic parabolic, hyperbolic equations in bounded domain and the whole space, the existence and the asymptotic behavior of solutions of stochastic evolution equations in Hilbert space, stochastic nonlinear partial differential equations driven by Brownian motion and levy process, the abstract theorems on existence and the long time behavior of solutions of these SPDEs. Meanwhile, this course will be exclusively devoted to stochastic partial differential equations of Ito type. We will employ the familiar tools such as the methods of eigenfunction expansions, the Green’s functions and Fourier transforms, together with the conventional techniques in stochastic analysis. The abstract theorems on existence, uniqueness and regularity of solutions will be proved and applied later to study the asymptotic behavior of solutions. The purpose of this course is to ensure students to master basic methods and theories in studying the solution properties of stochastic partial differential equations.
The content of this course includes:

• preliminaries such as Brownian Motions, martingales, Ito foumula and B-D-G inequality
• the existence, uniqueness and regularity properties of solutions of stochastic parabolic equations in bounded domain and the whole space respectively
• the analyasis of stochastic hyperbolic equations by adopting a similar approach
• the existence and uniqueness theorems for linear and nonlinear stochastic evolution equations for two kinds of solutions: the mild solution and the strong solution
• the boundedness, stability and the existence of invariant measures for the solutions
• several examples arising from physical models to show that the theorems given above has a wider range of applications
• a brief exposition on the connection between the stochastic PDEs and diffusion equations in infinite dimensions

Theory on Statistical Inference
Theory on Statistical Inference is the theory of drawing conclusions about populations or scientific truths from data. There are many modes of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Furthermore, there are broad theories and numerous complexities for performing inference. As one of basic compulsory courses, this course mainly talks about basic theories and methods on mathematical statistics and various statistical models, which ensures students to master basic theories and methods on mathematical statistics and apply these theories and methods skillfully into statistics models.
The content of this course includes two parts below. In the first part, we introduce basic theories and methods on mathematical statistics, mainly including point estimation, hypothesis testing, various types of principals and methods on optimal estimation and testing, decision theory, Bayesian theory, large sample and asymptotic theory, non-parameter inference and the method of sampling. In the second part, we introduce linear, generalized linear, nonlinear and non-parameter statistical models, mainly including the analysis of modern regression and variance and covariance, the diagnosis choice and inference on various kinds of parameter or non-parameter model, the theory of multivariate statistics distribution, data analysis method, the theory of modern testing design, modeling analysis and time series analysis.

Time Series Analysis
Time Series Analysisis an important branch of applied fields of probability and statistics, and has a wide application in a variety of fields, ranging from finance and economics, meteorology and hydrology to signal handle and mechanics. The main purpose of this course is to provide students with a rigorous theoretical foundation and empirical analysis skills to pursue applied projects involving economic and financial time series data, such as business applications (e.g., skillful usage of computer software packages) and research projects. The course focuses empirically and theoretically on time series methods that have become popular and are widely used in applied economics and finance.
The content of this course mainly includes characteristics of time series, univariate stationary time series models, principles of forecasting, estimation and inference in stationary ARMA models, vector autoregressive models, cointegration, unit root processes, nonlinear time series models and so on. Meanwhile, this course also provides a detail introduction to the frontier of time series analysis to help students find suitable research topics.

Econometric Models and Applications
Econometric Models and Application is based on the economics and mathematics statistics. It takes economic relations which have randomness characteristics as the research object, uses the mathematical model method to describe the relationship between specific economic variables, and provides specialized guidance theory and analysis method for economic analysis. The purpose of this course is to let students learn about the advanced statistical methods and corresponding application of statistical software, and the course also trains the students to use modern econometric analysis method to analyze the economic structure, the forecast of economic development, economic policy evaluation, etc.
Content of this course mainly includes five parts. The first part is the introduction to the relationship between econometrics and other disciplines and the research steps; the second part talks about the classical linear regression model, mainly introduces the classical linear regression model assumption, estimation, test and application; the third part introduces the ARCH model, including the test for correlation and heteroscedasticity; the fourth part analyzes the co-integration and error correction model; the last part introduces the panel data model. The course also explains the use of STATA and other statistical softwares.

Kernel Methods for Pattern Analysis
Kernel Methods for Pattern Analysisare a class of algorithms for pattern analysis in computer science, whose best known element is the support vector machine (SVM). It is known to us that the general task of pattern analysis is to find general types of relations in general types of data. It is widely used in neural network, statistic pattern recognition, machine learning and data mining, bioinformatics, document retrieval, etc. Kernel methods (KMs) find general types of relations by mapping the data into a high dimensional feature space, where each coordinate corresponds to one feature of the data items. Since the mapping can be quite general, the relations found in this way are accordingly very general. This course mainly describes a general framework for KMs applications. The main purpose of this course requires students to master basic theory on kernel method and its effective application, improve their design ability of non-linear classifier.
The main content of this course includes five parts. The first part is the main theoretical basis; the second part talks about several kernel based algorithms; the third part studies system from the simple to the more complex, e.g., Kernel partial least squares, Canonical correlation analysis, Support vector machine, Principal component analysis, and so on. Meanwhile, this course introduces several kernel functions, such as the basic examples, higher recursive kernel function, the kernel function generation model derived (such as HMM), the matching kernel function based on dynamic programming string, the special kernel function for processing text document and so on.

4. Supervisors

Quanxin Zhu, Hongjun Gao, Fengchang Xie, Zhibin Liang, Ming Yang, Zhen Pan.