Stochastic Partial Differential Equation Stochastic Partial Differential Equation, which emerged in s, 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. 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. 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.
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.
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. The inverse of this operator can be expressed in terms of X itself using the resolvent operator. This illustrates one of the connections between stochastic analysis and the study of partial differential equations.
An invariant measure is comparatively easy to compute when the process X is a stochastic gradient flow of the form. Thus, F is an H -function for the X -dynamics. Consider the Ornstein-Uhlenbeck process X on R n satisfying the stochastic differential equation. Dynkin's formula can be used to calculate many useful statistics of stopping times.
Dynkin's formula provides information about the behaviour of X at a fairly general stopping time. For more information on the distribution of X at a hitting time , one can study the harmonic measure of the process. That is, one wishes to study the first exit time.
Sometimes, however, one also wishes to know the distribution of the points at which X exits the set. The mean value property is very useful in the solution of partial differential equations using stochastic processes. Intuitively, the Green measure of a Borel set H is the expected length of time that X stays in H before it leaves the domain D. In particular, if the support of f is compactly embedded in D ,. From Wikipedia, the free encyclopedia.
This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. July Learn how and when to remove this template message. Main article: Continuous stochastic process. Main article: Markov property. Main article: Infinitesimal generator stochastic processes. Main articles: Kolmogorov backward equation and Fokker—Planck equation.
Main article: Feynman—Kac formula. Main article: Dynkin's formula. Main article: Harmonic measure. Main article: Green measure.
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