# STOCHASTIC PROCESSES

In probability theory, random phenomena that result from processes governed by probabilistic laws (such as the growth of a bacterial colony or the fluctuation of electric current in a circuit) are stochastic processes, according to Emanuel Parzen in Stochastic Processes. From a mathematical perspective, stochastic processes are collections of random variables and they involve events or phenomena with random outcomes. Random variables refer to variables whose values (e.g., true or false, yes or no) are determined by probabilistic laws. Stochastic processes include one or more random variables observed in multiple steps, or iterations, whose outcomes are not independent. In other words, the outcomes, while random, are related and can indicate a pattern of behavior. Consequently, stochastic processes can help eliminate some of the uncertainty associated with achieving various goals, because they take randomness into consideration.

Stochastic processes are commonly used in game theory examples, polling, tracking, probability calculations, and statistical analysis. In each case and at every step, an outcome may depend on any one of several random factors. Stochastic processes most often come into play for analyzing random yet predictable phenomena such as those of meteorology, business management, engineering, biology, and medicine.

Because of the existence of at least one random variable in stochastic processes, outcomes can never be predicted for certain according to deterministic laws. But because outcomes tend to indicate patterns of behavior, stochastic processes can yield knowledge in the form of predictions and they can indicate parameters of likely outcomes.

Stochastic processes may involve studying and measuring outcomes and patterns over a period of time for continuous phenomena. In a study involving the value of a stock, for example, a given variable A (which could represent a factor such as low unemployment) may be observed to increase value in 75 percent of past iterations. The variable B (which could represent a factor such as high unemployment) may be observed to decrease value in 75 percent of the cases. As a result, when A occurs, the observer may assume a 75 percent likelihood that value will increase.

The solution is not certain, because there is still a 25 percent chance that value will decrease. But past occurrences indicate a pattern of behavior suggesting that value is likely to increase when A is present and decrease when B is present. Repeated occurrences of variables and results may provide additional precision to the prediction, or identify the effects of other variables.

For single shot events, stochastic processes involve the measurement of a host of factors that can affect the outcome before and after each occurrence. Consider, for example, shooting a projectile toward a target 800 yards away. Measurement must be taken for and assumptions must be made about the weight of the projectile, amount of launch explosive, trajectory and direction of launch, wind direction and speeds, and even humidity.

If a single shot overshoots its target by 100 yards, adjustments must be made to gain accuracy. If additional shots are possible, variables such as trajectory, direction, and launch explosive may be adjusted to gain accuracy. The cycle is repeated on a third shot, then the fourth, until the target is hit.

Independent variables, such as atmospheric conditions, may change during the course of firings. They remain independent because changing wind speeds cannot be predicted with total accuracy. As a result, the projectile may be brought to land closer to the target, but unless conditions remain static, complete accuracy cannot be attained.

Stochastic processes can play an important role in managing a business and they have been applied to a few aspects of managing business operations, including sales strategies and inventory control. Stochastic processes in business work in a similar fashion to other fields. First, if a sales strategy for a product fails to produce an intended outcome, dependent variables may be adjusted. But independent variables, governed by the actions of competitors, remain unpredictable. This is especially relevant because competitors may alter their own actions based on the results of the first strategy.

Second, retailers, wholesalers, and manufacturers face two major inventory control problems: determining when to order additional stock items and determining how many to order. To operate one of these businesses efficiently, managers must consider the uncertain amount of items that will be needed in a given period as well as the uncertain amount of time it will take to have the items delivered after they are ordered. Because of these two uncertainties, companies must maintain an inventory of items so that they have them when they need them. Since having an inventory requires significant investment, companies strive to keep their inventories to a minimum to reduce overhead. Stochastic process would involve identifying the current policies on when to order and how much to order. If these policies are not successful—if a company frequently has too much or not enough inventory—then the company would have to alter them and reevaluate them until they yielded a more favorable outcome. This reevaluation might include consideration of seasonal fluctuations, changes in product movement because of economic conditions, demand levels for the different days of the week, and so on until a more precise picture of these factors had been obtained.

[ John Simley ,

updated by Karl Hell ]

Biswas, Suddhendu. Applied Stochastic Processes. New York: John Wiley & Sons, Inc. 1995.

Chou, Ya-lun. Statistical Analysis for Business and Economics. Amsterdam: Elsevier Science Publishing, 1989.

Johnson, Robert R. Elementary Statistics. 3rd ed. North Scituate, MA: Duxbury Press, 1980.

Parzen, Emanuel. Stochastic Processes. San Francisco: HoldenDay, 1962.

Wang, Yunzeng, and Yigal Gerchak. "Periodic Review Production Models with Variable Capacity, Random Yield, and Uncertain Demand." Management Science, January 1996, 130.