Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. Special Topics in Financial Mathematics. A basic knowledge of probability and statistics as well as transform methods for solving PDEs is assumed. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling.
Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Topics covered will be selected from standard options, exotic options, American derivative securities, term-structure models, and jump processes. Mathematical Finance'.
Prerequisites: Good knowledge of probability theory and differential equations. Some familiarity with analysis and measure theory is helpful.
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A course on pricing financial derivatives, risk management, and optimal portfolio selection using mathematical models. Students will be introduced to methods of Stochastic, Ito Calculus for models driven by Brownian motion. Models with jumps will also be discussed. Instructor: Cvitanic.
In addition, in looking through Caltech's current course offerings, it appears that there are several courses in statistics and stochastic systems that might benefit from better integration. These include:. Catalog listing Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion.
The course develops applications in selected areas such as signal processing Wiener filter , information theory, genetics, queuing and waiting line theory, and finance. The following table lists all of the courses that I was able to find that have been taught in the last four years. Enrollments when given are for , based on data from the registrar. The course listings below are from the Caltech catalog, mainly to serve as a reference for the rest of the information on this page. Introduction to Stochastic Processes and Modeling.
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Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Methods in Applied Statistics and Data Analysis. Prerequisite: Ma 2 or another introductory course in probability and statistics. Introduction to fundamental ideas and techniques of statistical modeling, with an emphasis on conceptual understanding and on the analysis of real data sets.
Multiple regression: estimation, inference, model selection, model checking. Regularization of ill-posed and rank-deficient regression problems. Principal component analysis. Discriminant analysis.
Stochastic Processes and Related Topics
Resampling methods and the bootstrap. Not offered — Instructor: Hill. Spacecraft Navigation. Prerequisite: CDS a. This course will survey all aspects of modern spacecraft navigation, including astrodynamics, tracking systems for both low-Earth and deep-space applications including the Global Positioning System and the Deep Space Network observables , and the statistical orbit determination problem in both the batch and sequential Kalman filter implementations. The course will describe some of the scientific applications directly derived from precision orbital knowledge, such as planetary gravity field and topography modeling.
Numerous examples drawn from actual missions as navigated at JPL will be discussed. Introductory Control Theory. An introduction to analysis and design of feedback control systems, including classical control theory in the time and frequency domain. Modeling of physical, biological, and information systems using linear and nonlinear differential equations. The ideas of organizing the school are: To establish cooperation and collaboration between participants and institution in South East Asian countries.
To popularize the subject of stochastic analysis which is an extremely useful branches of mathematics with huge range of applications. To give experiences to participants students from South East Asian countries for learning special topics in mathematics from experts in the field. Classification of states. Transition graphs. Topic 5. Occupancy times. Recurrent and transient states in terms of occupancy times.
Limit distribution. Stationary distribution. Occupancy distribution. Conditions for existence and uniqueness. Reducible chains. First-passage times. Topic 6. Counting processes. The exponential distribution as a model for waiting times. Memoryless properties.
Hazard rate. Minimum, maximum and sum of independent exponential random variables. The Poisson process as a counting process. The Poisson process as a process with stationary and independent increments. Conditioning to the total number of events. Compound Poisson processes. Topic 7. Time dependent transition probabilities. Transient analysis. Auxiliary discrete-time Markov chain.
List of stochastic processes topics
Topic 8. Renewal processes. Cumulative processes. Semi-Markov processes. Long-term analysis.