As with usual random variables, the state space \(\mathcal S\) can be discrete or continuous. However, if we want to track how the number of claims changes over the course of the year 2021, we will need to use a stochastic process (or “random process”).Ī stochastic process, which we will usually write as \((X_n)\), is an indexed sequence of random variables that are (usually) dependent on each other.Įach random variable \(X_n\) takes a value in a state space \(\mathcal S\) which is the set of possible values for the process. If we want to model, for example, the total number of claims to an insurance company in the whole of 2020, we can use a random variable \(X\) to model this – perhaps a Poisson distribution with an appropriate mean. Lots of the applications we will consider come from financial mathematics and actuarial science where the use of models that take into account uncertainty is very important, but the principles apply in many areas. In this module we will see many examples of stochastic models. For Apple shares, the price changes from day to day are highly uncertain, so a random model can account for the variability and unpredictability in a useful way.
Another word for a random model is a stochastic (“ sto- kass- tik”) model. To design a model requires a set of assumptions about how it will work and suitable parameters need to be determined, perhaps based on past collected data.Īn important distinction is between deterministic models and random models. Models allow us to try to understand and predict what might happen in the real world in a low risk, cost effective and fast way. For example, you might want to have a model to imitate the world’s population, the level of water in a reservoir, cashflows of a pension scheme, or the price of a stock. 22 End of Part II: Continuous time Markov jump processesĪ model is an imitation of a real-world system.20 Long-term behaviour of Markov jump processes.19.3 Hitting probabilities and expected hitting times.18.2 Transition semigroup and the forward and backward equations.18.1 Transitions in infinitesimal time periods.17 Continuous time Markov jump processes.16.2 Time inhomogeneous Poisson process.15.3 Forward equations and proof of equivalence.15.2 Example: sum of two Poisson processes.15.1 Definition 3: increments in infinitesimal time.15 Poisson process in infinitesimal time periods.14.3 Markov property in continuous time.14.2 Definition 2: exponential holding times.14 Poisson process with exponential holding times.13.3 Summed and marked Poisson processes.13 Poisson process with Poisson increments.Part II: Continuous time Markov jump processes.12 End of of Part I: Discrete time Markov chains.11.4 Proofs of the limit and ergodic theorems.11.2 Examples of convergence and non-convergence.11 Long-term behaviour of Markov chains.10.1 Definition of stationary distribution.8.3 Hitting and return times for the simple random walk.8.1 Hitting probabilities and expected hitting times.
6.3 A no-claims discount model with memory.5.1 Time homogeneous discrete time Markov chains.4.4 Expected duration for the gambler’s ruin.
4.3 Inhomogeneous linear difference equations.4.2 Probability of ruin for the gambler’s ruin.4.1 Homogeneous linear difference equations.2.3 Exact distribution of the simple random walk.1 Stochastic processes and the Markov property.
0 kommentar(er)
