Stochastic Processes

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• State the Kolmogorov equations for a Markov process where the transition intensities depend not only on age/time, but also on the duration of stay in one or more states
• Determine the stationary and equilibrium distributions of a Markov chain
• Classify the states of a Markov chain as transient, null, recurrent, positive recurrent, periodic, aperiodic and Ergodic
• Demonstrate how a Markov chain can be simulated
• Recall the definition and derive some basic properties of a Poisson process
• Define and explain the basic properties of Brownian motion, demonstrate an understanding of stochastic differential equations and then to integrate, and apply the Ito formula
• Calculate the distribution of a Markov chain at a given time
• Classify a stochastic process according to whether it operates in continuous or discrete time and whether it has a continuous or a discrete state space, and give examples of each type of process
• Derive the Kolmogorov equations for a Markov process with time independent and time/age dependent transition intensities
• Solve the Kolmogorov equations in simple cases
• Understand the definition of a stochastic process and in particular a Markov process, a counting process and a random walk
• Understand, in general terms, the principles of stochastic modelling
• Understand survival, sickness and marriage models in terms of Markov processes
• Demonstrate how a Markov jump process can be simulated
• Describe a time-inhomogeneous Markov chain and its simple applications
• Describe a Markov chain and its transition matrix

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.