Stochastic Processes

Motivating Stochasticity

• A stochastic process is a sequence of random variables
• A stochastic process is indexed by some other variable
• A stochastic process is typically index by time
• We can define a stochastic process as $S$, which can also be thought of as a sequence of successive random variables:

$$S = S_{0}, S_{1}, S_{2}, ..., S_{t}, ..., S_{n}$$

• These successive random variables all belong to the same function $S$
• In other words, the space each of these random variables lives over is the same, and when we need to talk about that space, then we’ll talk about $S$, and any realizations of $S$ will be written as $s$

Difference between Stochasticity and Determinism

• Determinism is any process that isn't stochastic, or doesn't involve an element of randomness
• In a stochastic process, we are interested in a best guess
• In a deterministic process, we are interested in a consistent solution
• A deterministic process can be determined by an exact formula, whereas a stochastic process can't be determined by an exact formula and involves some guessing

Examples of Deterministic Processes

• We only need to know an object's mass and acceleration in order to determine the force of an object

$$F = ma$$

• We only need to know the degrees in Celsius in order to determine the degrees in Kelvin

$$K = C + 273.15$$

Examples of Stochastic Processes

• Rolling a die
• Flipping a coin

• Rolling a die based on last flip

  • This is an example of a Markovian Chain, which has the same probability space as rolling a die (without knowing the last toss)

• Flipping a coin (based on last flip)

  • This is an example of a Markovian Chain, which has the same probability space as flipping a coin (without knowing the last flip)

References

• Probability, Statistics and Stochastic Processes
• Example of Stochasticity
• Description of Stochasticity in Markov Chains
• Lecture Notes about Stochastic Processes