## Stochastic Processes

There are two types of actuarial models that are commonly used to create scenarios that model future contingent events; they are stochastic and deterministic models. Deterministic models are built with constant values and therefore are not effective in projecting scenarios that are associated with random variables. Stochastic models and their associated processes are becoming popular with actuaries because the processes estimate a probability distribution in their outcomes. Stochastic processes use random variables as inputs and take into account the variable impact on the outcome over time. Stochastic processes are more complex to model than deterministic processes, but because they provide the user with quantitative data about the probability of scenarios, they are invaluable in the assessment of complex risk scenarios. Stochastic processes have proven to be valuable tools in modeling catastrophic risk which allow actuaries to develop scenarios about "what could happen" rather than just looking at historical data and what did happen. This essay investigates some of the business applications that are evolving around the use of stochastic processes. This article also discusses how stochastic models and processes are helping to project the probability of certain scenarios regarding the solvency of the U.S. Social Security system. Users of stochastic processes are able to evaluate a large number of possible outcomes having to do with a number of potential risks. The real value added that stochastic processes provide to users today is the ability to not only model complex risk scenarios, but also to assess the likelihood of the risk actually occurring.

Keywords Actuarial Uncertainty; Catastrophic Risk Models; Deterministic Model; Probabilistic Models; Random Variables

### Actuarial Science: Stochastic Processes

OverviewIn general, when discussing actuarial models, there are two distinct types of models that are used: Deterministic models and stochastic models. This article focuses on the advantages of using stochastic or probabilistic models to model risk scenarios. This topic would be incomplete if one were not to consider the varied reasons why stochastic models are replacing deterministic models in the areas of insurance and finance. Early deterministic models centered upon examining what the results of past events would look like if the event were to happen today (Boyle, 2002). The major drawback in using "a deterministic model is… that [it] takes no account of random variation and therefore gives a fixed and precisely reproducible result" ("Modeling biological systems," n.d.).

Deterministic models are not without their complexity; "deterministic models are often described by sets of differential equations, […] and also rely on numerical analysis and computer simulation" in their development. It is not the lack of complexity that puts deterministic models at a disadvantage for modeling today's complex risks, but rather that "deterministic models find statistical variations in the average behavior of the system that are relatively unimportant" ("Modeling biological systems," n.d.). To reiterate, deterministic models don't rely upon random events (random variables) in modeling their outcomes. According to Uday Virkud of Applied Insurance Research, "History is too short for an actual model to see what is possible" (Boyle, 2002). Since deterministic models rely on historical experience and look only at what has actually happened in the past, deterministic models don't adequately assess what events or outcomes could actually happen in the future (Boyle, 2002).

The following description of a stochastic model is taken from its use in the biological sciences, but explains nicely the importance of random events and random times as inputs into the probabilistic model. "Stochastic models should be used […] where there is reason to expect random events to have an important influence on the behavior of the system. It may also be necessary to take account of events occurring at random times. The essential difference between a stochastic and deterministic model is that in a stochastic model, different outcomes can result from the same initial conditions" ("Modeling biological systems," n.d.).

The nature and complexity of risk in today's insurance and financial markets requires that actuarial models keep pace in modeling complex risk scenarios. Stochastic models have been in use for several decades, but became popular initially as a means to model catastrophic risk. Stochastic models are needed to model complex catastrophic models because they meet the following three imperatives:

- They model the chances of an event happening.
- They help to define the upper and lower limits of a catastrophic (event) occurrence.
- Depending upon the intensity, these models can project what effect the event will have (in a given area).

Stochastic or probabilistic models don't predict when an event may occur, but instead use mathematical probabilities to help insurers and the insured assess what will happen if a given event or situation were to occur (Boyle, 2002).

There have been a number of 'defining' events in the insurance industry in the past 15 years that have reinforced the value of using stochastic models. Stochastic models and their use of random events provide a much more comprehensive view of possible risk outcomes. One industry analyst stated that prior to Hurricane Andrew in 1992, "no one could fathom a hurricane that could cost $13 billion" (Boyle, 2002). Suddenly, there was a realization that past experience was not enough to model risk; the industry needed to examine what a single event like Hurricane Andrew could do. Insurers quickly realized that stochastic models were the best option for actually gauging their risk.

Stochastic models incorporate data from many different sources, including: Census data, demographic data and past history of events. Actuaries who build stochastic models don't limit the data sources for their models; any and all data that may be relevant in determining actual risk is considered for use in the model. Stochastic models rely heavily upon variables that are site specific; local conditions regarding geology, urbanization and local weather hazards are all customized in the stochastic modeling process. Stochastic models provide valuable information to a number of users; local insurance agents, insurance companies, re-insurers, banks and corporate clients all benefit from the output of stochastic models.

Future Trends in Stochastic ModelingThere are a number of trends occurring in the use of stochastic models today. Third party vendors such as Risk Management Solutions (RMS) develop stochastic models that are rented to insurers to help with risk assessment. It is acknowledged that using more than one model helps to identify more possible risks and thus allows insurers to further diversify their risks. Not only is it important for insurers to use more than one model for risk assessment, but stochastic models cannot remain static. Stochastic models need to be upgraded constantly with new data and incorrect assumptions (variables) must be modified to improve the scenario outcomes.

Dynamic stochastic models incorporate the systematic process for revisiting the model in response to observed results. Models chart plausible future outcomes within a given framework. Stochastic models provide a range of possible future outcomes that in totality imply something about a reasonable range in which future actual results can be expected to lie ("The roles," 2006).

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