Comparison Methods For Stochastic Models And Risks
There are two basic kinds of models: The first kind are deterministic models and the second kind are stochastic, or probabilistic models. There are significant differences between them, and both types are useful in the the business world. It is important, however, to understand how they are different, and how those differences impact the choices a modeler or analyst makes when trying to decide which kind of model to use to answer a particular question.
A deterministic model is essentially a formula. If you always start with condition A and condition B you will always get result C. Every.Single.Time. Consider the most commonly quoted mathematical equation: 2+2=4. This model has two inputs. The first number and the second number. The model executes an mathematical operation and returns the answer. Addition being what it is, if the inputs to this model are always 2 and 2, the output will always be 4.
Univariate Stochastic Orders Theory of Integral Stochastic Orders Multivariate Stochastic Orders Stochastic Models, Comparison and Monotonicity Monotonicity and Comparability of Stochastic Processes Monotonicity Properties and Bounds for Queueing Systems Applications to Various Stochastic Models Comparing Risks. List of Symbols. We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals. Stochastic orders are important approximation tools that provide valuable insight into the behaviour of complex stochastic models. Research into stochastic orders is blossoming, with many open problems being studied and a wide range of applications explored.
What Is Stochastic Modeling
This principle of Starting Condition A and Starting Condition B yield Result C scales upward into very complex, very large models. However, if those complex models are deterministic at their core they will always return the same result if the starting conditions do not vary. This property of deterministic models makes them very useful for determining whether to make a choice between two or more discrete options. For example, consider the possibility that Southwest Airlines is considering investing in an entirely new fleet of aircraft that will be usable until 2050.
Currently, Southwest's fleet consists entirely of Boeing 737s. This allows Southwest to streamline its maintenance and other procedures. Southwest has a great deal of data about it's operations, and can build a model that accurately reflects the cost of operating. If it wants to add a route, it knows exactly how much fuel will be required and how much the maintenance costs will be. By assigning values (which it knows from historical data) to different operating parameters, Southwest can evaluate a potential change in aircraft type by substituting different flight and operating parameter values for each possible aircraft. When the model is run and the results recorded, Southwest can directly compare the costs of operating its current fleet to the cost of operating a different aircraft.
The second general type of models are stochastic models. Stochastic models incorporate one or more probabilistic elements into the model, which means that the final output of the model will typically be some kind of confidence interval with a most-likely point estimate. A very simple stochastic model might be rand() + 2 . The rand() input will return (if you do this in Excel) a random number between 0 and 1. Every time you run this model you'll get a different answer between 2 and 3. Here's proof.
Something to remember is that all random numbers are not the same. In Excel, the rand() function returns a random number based on the familiar Normal or Gaussian Distribution. If the event that you are trying to model is actually normally distributed, this isn't a problem. But many events in our daily lives aren't normally distributed. For instance, if you are trying to model the drive thru lane at Burger King, the number of cars arriving each minute isn't normally distributed. Arrivals follow a Poisson Distribution. The amount of time between each arrival follows an Exponential Distribution. These are beyond the scope of this article, but it's important to remember that the model builder needs to ensure that the event being modeled is accurately reflected in the rand() statement.
What are stochastic models used for? There a lots of uses for a stochastic model, but in general the all come down to trying to accurately portray the likelihood of an event or a series of events occurring. Risk management and mitigation is one area that uses stochastic modeling. A simple risk model is (probability of an event) x (cost of the event). A concert promoter may want to know what the cost at risk for a cancelled concert series is. Each concert has 1000 tickets sold at $100 each. If a concert is cancelled she will have to refund those tickets at a cost of $100,000. The weatherman says there is a 10% chance of rain. A simple risk assessment says that she has $10,000 at risk in this scenario.
In my next post I'll examine a specific kind of stochastic model...Monte Carlo simulation. A Monte Carlo simulation is used to model the potential outcomes of a series of stochastic events. For example, who would win a notional 8-team college football playoff this year? Come back next week and we'll get into the mechanics of stochastic modeling.