The exhaustive list of topics in Stochastic Processes in which we provide Help with Homework Assignment and Help with Project is as follows:
- Probability Theory Refresher: Axiomatic construction of probability spaces, random variables and vectors, probability distributions, functions of random variables; mathematical expectations, transforms and generating functions, modes of convergence of sequences of random variables, laws of large numbers, central limit theorem.
- Introduction to Stochastic Processes (SPs): Definition and examples of SPs, classification of random processes according to state space and parameter space, types of SPs, elementary problems.
- Discrete-time Markov Chains (MCs): Definition and examples of MCs, transition probability matrix, Chapman-Kolmogorov equations; calculation of n-step transition probabilities, limiting probabilities, classification of states, ergodicity, stationary distribution, transient MC; random walk and gambler’s ruin problem, applications.
- Continuous-time Markov Chains (MCs): Kolmogorov- Feller differential equations, infinitesimal generator, Poisson process, birth-death process, Applications to queueing theory, inventory analysis, communication networks, finance and biology.
- Brownian Motion: Wiener process as a limit of random walk; first -passage time and other problems, applications to finance.
- Branching Processes: Definition and examples branching processes, probability generating function, mean and variance, Galton-Watson branching process, probability of extinction.
- Renewal Processes: Renewal function and its properties, elementary and key renewal theorems, cost/rewards associated with renewals, Markov renewal and regenerative processes, applications.
- Stationary Processes: Weakly stationary and strongly stationary processes, moving average and auto regressive processes.
- Martingales: Conditional expectations, definition and examples of martingales, inequality, convergence and smoothing properties, applications in finance.