Stochastic: Variable Determined by Chance

An in-depth exploration of stochastic processes, concepts, and applications in various fields like statistics, regression analysis, and technical securities analysis.

A stochastic variable, also known as a random variable, is a variable whose value is subject to fluctuations influenced by random factors. Unlike deterministic variables, which have fixed outcomes, stochastic variables are inherently unpredictable and can be characterized by probability distributions.

Types of Stochastic Processes

Discrete Stochastic Processes

Discrete stochastic processes consist of a sequence of random variables indexed by discrete time points. An example is the Poisson process, which models events occurring randomly over time.

$$ X_n \text{ for each } n \in \mathbb{N} $$

Continuous Stochastic Processes

Continuous stochastic processes describe variables that evolve over a continuous time period. An example is the Wiener process, fundamental in financial mathematics for modeling stock prices.

$$ X_t \text{, where } t \in [0, \infty)$$

Stochastic Variables in Regression Analysis

In regression analysis, the dependent variable is often stochastic. This means that even though an explanatory model is used, not all factors affecting the dependent variable can be perfectly predicted or accounted for by the model.

Ordinary Least Squares (OLS) Regression

In OLS regression, the dependent variable (\(Y\)) is expressed as:

$$ Y = \beta_0 + \beta_1X_1 + ... + \beta_nX_n + \epsilon $$

Here, \(\epsilon\) (error term) represents the stochastic component of the dependent variable.

Example

Given a linear model \(Y = 2 + 3X + \epsilon\),

• \(Y\) is the stochastic dependent variable.
• \(X\) is the independent variable.
• \(\epsilon\) captures randomness or unexplained variability.

Stochastics in Technical Securities Analysis

Stochastics is a critical branch of technical analysis used in financial markets to predict future price movements based on historical data.

Stochastic Oscillator

A widely used indicator in technical analysis, the stochastic oscillator compares a particular closing price of a security to its price range over a specified period. It’s formulated as:

$$ \text{Stochastic \%K} = \frac{(C - L_{14})}{(H_{14} - L_{14})} \times 100 $$

Where:

• \(C\) is the most recent closing price.
• \(L_{14}\) and \(H_{14}\) are the lowest and highest prices in the past 14 periods.

Historical Context

The term “stochastic” originates from the Greek word “stochastikos,” meaning “pertaining to conjecture” or “random.” Its formal application in statistical mathematics has evolved significantly, particularly in areas like quantum mechanics, econometrics, and machine learning.

Special Considerations

Understanding and working with stochastic variables require:

• Knowledge of probability theory.
• Competence in constructing and interpreting various probability distributions.
• Proficiency in statistical software tools for simulation and modeling.

Comparisons with Deterministic Models

• Deterministic Models: Predict outcomes with certainty given initial conditions.
• Stochastic Models: Include random variation and uncertainty, better representing real-world complexities.

Related Terms

• Random Variable: A variable that assumes different values due to random phenomena.
• Probability Distribution: A mathematical function that describes the likelihood of different outcomes.
• Stochastic Differential Equations (SDEs): Equations used to model systems influenced by random noise.

FAQs

What is a stochastic process?

A stochastic process is a collection of random variables representing the evolution of some system of random values over time.

How are stochastic methods used in finance?

Stochastic methods are used to model and forecast stock prices, interest rates, and other financial metrics by incorporating random variability.

Why is the dependent variable viewed as stochastic in regression?

Because there are always unknown variables and inherent randomness affecting the outcome, making perfect prediction impossible.

References

• Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
• Fama, E. F. (1965). “The behavior of stock-market prices”. Journal of Business, 38(1), 34-105.
• Monte, F. (2001). Stochastic Modeling for Real Estate Valuation.

Summary

Stochastic variables play a vital role in many fields, from finance to physics, due to their ability to model systems influenced by randomness. Whether in regression analysis or securities analysis, understanding and applying stochastic principles allow for more robust and realistic models, accommodating the inherent uncertainties of the real world.