Content The first half of the course will cover the basics of Probability Theory while the second half will delve into the theory of Stochastic Processes. Below is the list of topics that will be covered in the course. The mathematical representation of random phenomena: The probability space, properties of the probability measure, Independence of events, Conditional probability and Bayes formula, applications to parameter inference.
An Introduction to Continuous-Time Stochastic Processes
Random Variables and their distributions: Discrete and continuous random variables, Expectation and Variance, Important Examples of Random Variables, Independent random variables and their sums, Conditional Distribution and Conditional Expectation, Markov and Chebyshev inequalities. Law of total variation, estimation of intrinsic and extrinsic noise in biological systems.
Convergence of Random Variables: Modes of convergence, Laws of large numbers, the central limit theorem, the law of the iterated logarithm, Applications to the analysis of cell population data. Generating functions and their applications: Definition and important examples, Random Walks, Branching processes, Coalescent processes, Modeling epidemic processes and stem-cell differentiation. Markov chains: Transition functions and related computations, Classification of states and classification of chains.
Concepts of recurrence, transience, irreducibility and periodicity, Stationary distributions, Continuous time Markov Chain model of a biochemical reaction network. Introduction to the theory of Martingales: Basic definitions, Martingale differences and Hoeffding's inequality, Martingale Convergence Theorem, Crossings and convergence, Stopping times and the optional sampling theorem, Doob's maximal inequalities, Applications to the analysis of stochastic biochemical reaction networks.
Literature While no specific textbook will be followed, much of the material and homework problems will be taken from the following books: An Introduction to Stochastic Processes with Applications to Biology, Linda Allen, Second Edition, Chapman and Hall, Students are expected to have a good grasp of Linear Algebra and Multivariable Calculus. We assume that the reader is already familiar with the basic motivations and notions of probability theory.
We commence along the lines of the founding work of Kolmogorov by regarding stochastic processes as a family of random variables defined on a probability space and thereby define a probability law on the set of trajectories of the process.
More specifically, stochastic processes generalize the notion of finite-dimensional vectors of random variables to the case of any family of random variables indexed in a general set T. The financial industry is one of the most influential driving forces behind the research into stochastic processes.
An Introduction to Continuous-Time Stochastic Processes | ehonahyjabim.tk
This is due to the fact that it relies on stochastic models for valuation and risk management. But perhaps more surprisingly, it was also one of the main drivers that led to their initial discovery. This chapter is devoted to an analysis of fundamental results related to topics that have attracted the attention of a large number of scientists from many disciplines.
The key issue is individual-based models and their approximation, leading to the so-called mean field models and to nonlinear PDEs.
This category includes ant colonies, herd behavior, and swarm intelligence, all of which have generated a large and current body of research in biology, physics, operations research, economics, and related fields. Print ISBN