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Stochastic Process is defined as a collection of random variables. Another name for Stochastic is random variables. The number of random variables depends on the variable parameters. It is widely used as a mathematical model and in probability theories. It is simply a probability process, in this we can analyze and evaluate the probability of any event. It’s a study of Random signals, independence, expectations, discrete time space models, minimum phase spectral factor, orthogonal functions, averages, spectral density, estimation of dynamical systems, Markov chains, context free grammars, Gaussian state space models, allocation in wireless communications, stochastic sensor scheduling, probability distribution, conditional probability etc.
Various important topics included in Stochastic Process are as follows:
- Stochastic State Space Model
- Bayesian Filtering
- Markov Processes
- Spectral analysis
- Levinson filter
- Kalman filter
Various processes involved in stochastic process are:
- Bernoulli process - It is defined as the sequence of finitely or infinitely distributed random variables, where each random variable takes either the value zero or value one. Let us consider the probability of head occurring when we toss a coin let it be p then the probability of tail is 1-p hence we can say that if probability of occurrence of head is p that is 1 then the probability of occurrence of tail will be 1-p that is 0.
- Random walk - It is defined as the total of all the random variables or random vectors in Euclidean space, hence they act as those processes which change their values in discrete interval of time. Here in order to form a lattice path the location can even jump or walk to its neighboring sites of lattice hence it’s called random walk because the location can jump or walk to its neighboring sites. It is also called as bordered symmetry random walk if its state space is limited to finite dimensions.
- Poisson process- It is defined as a counting process, which represents random number of points or events up to some time. It represents the number of points between the intervals from zero to a given time in the Poisson’s random variable. This process consists of natural numbers and non negative numbers such that the natural numbers acts as its state space and non negative numbers as its index set. We can also call it as Poisson’s counting process
Filtering problems is that model which is used for number of state estimation problems in signal processing and various other fields. We prefer linear filters for Gaussian random variables and are also known as Wiener filters. Sometimes filtering comes as a solution or part of solution of an optimal control problem. It works on the principle to establish the best approximate values from incomplete set of observations. Dynkin’s formula is a mathematical theorem which provides us an approximate value of any statistics. It is the generalized form of Calculus theorem. It offers a good class of functions. Brownian motion or Wiener process- The Wiener process was named after the scientist Norbert Wiener who discovered about the study of continuous-time stochastic process. This process is also as Brownian motion. It plays a vital role in pure and applied mathematics this is because in pure mathematics it gave rise to concepts of continuous-time martingales. While in applied mathematics it told about the white noise Gaussian process. It found its popularity in the concepts of mathematics and physics.
Levy process plays an important role in integral-differential equations. It is a random trajectory which deals with generalized concepts of Brownian motion. It may even contain some discontinuities. It’s a stochastic process having independent or stationary increments. It determines the motion of certain points whose successive displacements are random and independent.
Martingales- It plays a very vital role in stochastic process. In this process the expected value at each step is equal to its previous realized value. It consists of a primer whose purpose is to make the reader understand the concepts and significance of Martingales. In this a person wins his stake if heads comes up on coin and loses if tail comes up.
It consists of various advanced topics such as:
- Markov Processes and Time
- Blumenthal’s 0-1 law
- Markov Chains
- Poisson Processes
- Levy’s canonical form
- Stochastic solutions of problem
- Gaussian additive process
- Stochastic integral-differential equations
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- Models, Markov Chains, Gauss Markov Processes, Stochastic Context Free Grammars, Stochastic Simulation, Simulation based Stochastic Optimization, Recursive Least Squares to Discrete Optimization, Stochastic Approximation Algorithms and Analysis, Optimal Filtering, Kalman and Hidden Markov Model Filters, Sequential Markov Chain Monte-Carlo, Continuous-time Stochastic Filtering Theory, Asymptotic Properties of Maximum Likelihood Estimation, Expectation Maximization Algorithms, Hidden Markov Models, Gaussian State Space Models, Stochastic Context Free Grammars.
- Finite horizon and infinite horizon problems, Structural Results using super modularity, Resource Allocation in Wireless Communications, Partially Observed Stochastic Control, Stochastic Sensor Scheduling, Stochastic Simulation of Biological Ion Channels at Atomic Scale, Primarily on Brownian dynamics simulations, Structural Results in Stochastic Games, Applications in Sensor Networks, Game Theory is a natural extension of Markov Decision Processes, Intuitive Notion of Probability, Axiomatic Probability, Joint and Conditional Probability, Random Variables, Probability Distribution and Density Functions, Normal or Gaussian Random Variables, Impulsive Probability Density Functions, Multiple Random Variables
- Sum of Independent Random Variables and Tendency toward Normal Distribution, Transformation of Random Variables, Multivariate Normal Density Function, Linear Transformation and General Properties of Normal Random Variables, Concept of a Random Process, Probabilistic Description of a Random Process, Gaussian Random Process, Stationarity, Ergodicity, and Classification of Processes, Autocorrelation Function, Crosscorrelation Function, Power Spectral Density Function, Cross Spectral Density Function, White Noise, Random Telegraph Wave, Wiener or Brownian-Motion Process, Determination of Autocorrelation, Spectral Density Functions from Experimental Data, Steady-State Analysis, Integral Tables for Computing Mean-Square Value, Noise Equivalent Bandwidth, Shaping Filter, No stationary (Transient) Analysis, Forced Response, Optimization with Respect to a Parameter, The Stationary Optimization Problem, Weighting Function Approach, Orthogonality, Complementary Filter, Perspective, Estimation, Markov Processes.
- Vector Description of a Continuous-Time Random Process, Discrete-Time Model, The Discrete Kalman Filter, Scalar Kalman Filter Examples, Solution of the Matrix Riccati Equation, Divergence Problems, Complementary Filter Methodology, INS Error Models, Damping the Schuler Oscillation with External Velocity Reference Information.
Help for the complex topics like :
- Introduction and Definitions of Markov property, Class Structure, Absorption Probabilities, Strong Markov Property, Measure Theory, Recurrence and transience, Stationary distributions, Jump Processes, sample spaces, and events; axioms of probability, conditional probability and independence, Random Variables: Discrete random variables and probability distribution functions;, binomial, Poisson, and geometric distributions, Continuous random variables:, probability density and cumulative density functions;, uniform, exponential, and normal densities, Joint distributions; marginals, Random Variables + Expectations:, Calculations involving independent random variables, Expectations, variance, covariance, moments, and moment generating function, Markov Chains, Markov property: Definitions, Transition probabilities; hitting times, Transition matrix, State Space Structure: Recurrence and transience, Decomposition of state space, Application: Birth-death chains, Application: Queueing chains and branching process, Stationary distributions: Definition and basic properties, Average number of visits to a recurrent state, Null and positive recurrence and stationary distributions, Decomposing state space: Irreducible closed sets and stationary distributions;, linear algebra interpretation for finite-state chains, Convergence to stationary distribution: period, aperiodicity, and Perron-Frobenius theorem, Jump processes, Jump processes: Definitions and examples of jump processes, Markov jump processes, and Chapman-Kolmogorov equations, Basic theory: Forward and backward equations, Poisson process, Birth-death processes:, Properties of Markov jump processes.
- Recurrence, transience, and stationary distributions, Properties of birth-death processes, Queues, little renewal theory, Renewal equation, Residual and total life Discrete renewal processes, Second-order and gaussian processes, Wiener processs, Gaussian processes: Definitions and some examples, Stationary gaussian processes and linear time-invariant systems, Spectral power densities, filtering, Elements of probability theory, Stochastic processes: basic definitions, Discrete time Markov processes: Markov chains, Continuous time Markov processes, Example: Brownian motion, Diffusion processes: basic definitions, the generator, Backward Kolmogorov and the Fokker-Planck equations, Stochastic differential equations (SDEs); Ito calculus, Ito and Stratonovich stochastic integrals,, connection between SDEs and the Fokker-Planck equation, Applications, Bistability, metastability and exit problems, Stochastic mode elimination, Spatially extended systems, Markov Chain Monte Carlo (MCMC), Numerical methods for SDEs, Statistical mechanics, Markov processes, Random Walks, Martingales, Brownian motionIntroduction to stochastic integrals and differential equations, Applications.
- Discrete and continuous time,properties of discrete-parameter martingales,martingale theory,Foster-Lyapunov ,recurrence and speed of convergence,stochastic processes,reversibility and martingales,,Poisson processes,,Renewal processes,,Probability spaces,,countable additivity,Conditional expectation,change-of-variable and transforms of r.v.'s ,Spaces of trajectories ,Independent increments,stationarity and wide-sense stationarity,Markov property,ergodicity ,Multistep transition probabilities,Chapman-Kolmogorov equation. ,First-step analysis,Classification of states. ,Reducibility ,Recurrence,Steady state,Time reversibility & regeneration,Recurrence and Ergodicity,recurrence and positive recurrence,Random-walk & birth-death ,Empirical averages,Ergodic Theorem ,Renewal reward theorem,Proof of equilibrium,Coupling method of proof ,Regenerative processes,Martingales & Applications. ,Definitions and Optional Sampling Theorem. ,Expectation calculations in examples.,Random-walk,branching-process,Poisson Processes & Continuous-time Chains,Poisson process def'ns and characterizations ,Relation to Discrete-time chains. Embedding. ,Transition probabilities,Birth-death process examples,Analysis of Absorption,Queueing & Other Applications ,Queueing examples,Reversibility,Applications in Queueing & Markov Chain Monte Carlo.
- Semimartingales, stochastic integration, Ito's formula, Girsanov's theorem.
- Gaussian and related processes.
- Stationary/isotropic processes.
- Integral geometry and geometric probability.
- Maxima of random fields and applications to spatial statistics and imaging
- Introduction to measure theory
- Lp spaces and Hilbert spaces
- Random variables
- Conditional expectation
- Conditional distribution
- Uniform integrability
- Lp convergence
- Stochastic processes
- Sample path continuity
- andom walk
- Markov chains
- Gaussian processes
- Poisson processes
- Construction and basic properties of Brownian motion