🔀Stochastic Processes Unit 3 – Stochastic processes basics

Key Concepts and Definitions

• Stochastic process $\{X(t), t \in T\}$ is a collection of random variables indexed by a parameter $t$, often representing time
• State space $S$ is the set of all possible values that the random variables $X(t)$ can take
  • Can be discrete (finite or countably infinite) or continuous (uncountably infinite)
• Sample path or realization is a single possible trajectory of the stochastic process over time
• Stationarity implies that the joint probability distribution of the process does not change when shifted in time
  • Strictly stationary if the joint distribution is invariant under any time shift
  • Weakly stationary (or covariance stationary) if the mean and covariance remain constant over time
• Ergodicity suggests that the time average of a single realization converges to the ensemble average as the time interval grows
• Markov property states that the future state of the process depends only on the current state, not on the past states
  • $P(X(t_n) \leq x_n | X(t_{n-1}) = x_{n-1}, \ldots, X(t_1) = x_1) = P(X(t_n) \leq x_n | X(t_{n-1}) = x_{n-1})$

Types of Stochastic Processes

• Discrete-time processes have a countable index set $T$, often representing equally spaced time points (e.g., $T = \{0, 1, 2, \ldots\}$)
  • Examples include random walks and discrete-time Markov chains
• Continuous-time processes have an uncountable index set $T$, typically representing a continuous time interval (e.g., $T = [0, \infty)$)
  • Examples include Poisson processes and Brownian motion
• Markov processes satisfy the Markov property, where the future state depends only on the current state
  • Can be discrete-time (Markov chains) or continuous-time (continuous-time Markov processes)
• Gaussian processes have joint distributions that are multivariate normal
  • Characterized by a mean function and a covariance function
• Renewal processes consist of a sequence of independent and identically distributed (i.i.d.) random variables representing inter-arrival times
• Point processes describe the occurrence of events in time or space, with the most notable example being the Poisson process

Probability Theory Foundations

• Probability space $(\Omega, \mathcal{F}, P)$ consists of a sample space $\Omega$, a $\sigma$-algebra $\mathcal{F}$, and a probability measure $P$
• Random variable $X$ is a measurable function from the sample space $\Omega$ to the real numbers $\mathbb{R}$
• Cumulative distribution function (CDF) $F_X(x) = P(X \leq x)$ fully characterizes the distribution of a random variable $X$
• Probability mass function (PMF) for discrete random variables: $p_X(x) = P(X = x)$
• Probability density function (PDF) for continuous random variables: $f_X(x) = \frac{dF_X(x)}{dx}$
• Expected value $E[X] = \sum_{x} x p_X(x)$ for discrete $X$ and $E[X] = \int_{-\infty}^{\infty} x f_X(x) dx$ for continuous $X$
• Variance $\text{Var}(X) = E[(X - E[X])^2]$ measures the dispersion of a random variable around its mean
• Covariance $\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]$ quantifies the linear dependence between two random variables $X$ and $Y$

Markov Chains

• Markov chain is a discrete-time stochastic process satisfying the Markov property
  • State space $S$ can be finite or countably infinite
• Transition probability $p_{ij} = P(X_{n+1} = j | X_n = i)$ represents the probability of moving from state $i$ to state $j$ in one step
• Transition probability matrix $P = [p_{ij}]$ contains all one-step transition probabilities
  • Each row of $P$ sums to 1, as it represents a probability distribution
• $n$-step transition probability $p_{ij}^{(n)} = P(X_{m+n} = j | X_m = i)$ gives the probability of moving from state $i$ to state $j$ in $n$ steps
• Chapman-Kolmogorov equations relate $n$-step transition probabilities to one-step transition probabilities: $p_{ij}^{(m+n)} = \sum_{k \in S} p_{ik}^{(m)} p_{kj}^{(n)}$
• Stationary distribution $\pi$ satisfies $\pi P = \pi$ and represents the long-run proportion of time spent in each state
  • For irreducible and aperiodic Markov chains, the stationary distribution is unique and exists regardless of the initial state

Poisson Processes

• Poisson process $\{N(t), t \geq 0\}$ is a continuous-time counting process satisfying certain properties
  • $N(t)$ represents the number of events that have occurred up to time $t$
• Increments $N(t) - N(s)$ for $t > s$ are independent and Poisson distributed with mean $\lambda(t - s)$, where $\lambda > 0$ is the rate parameter
• Inter-arrival times between consecutive events are independent and exponentially distributed with mean $1/\lambda$
• Poisson processes are memoryless, meaning that the waiting time until the next event does not depend on the time since the last event
• Poisson processes have stationary and independent increments
  • Stationary increments: the distribution of $N(t) - N(s)$ depends only on the time difference $t - s$
  • Independent increments: for disjoint time intervals, the increments are independent random variables
• Poisson processes are often used to model the occurrence of rare events, such as customer arrivals or machine failures

Brownian Motion

• Brownian motion (or Wiener process) $\{B(t), t \geq 0\}$ is a continuous-time stochastic process with certain properties
  • $B(0) = 0$ almost surely
  • Increments $B(t) - B(s)$ for $t > s$ are independent and normally distributed with mean 0 and variance $t - s$
• Sample paths of Brownian motion are continuous but nowhere differentiable
• Brownian motion has stationary and independent increments
• Brownian motion is a Gaussian process with mean function $\mu(t) = 0$ and covariance function $\text{Cov}(B(s), B(t)) = \min(s, t)$
• Variations of Brownian motion include:
  • Drifted Brownian motion: $X(t) = \mu t + \sigma B(t)$, where $\mu$ is the drift parameter and $\sigma$ is the volatility parameter
  • Geometric Brownian motion: $S(t) = S(0) \exp(\mu t + \sigma B(t))$, often used to model stock prices
• Brownian motion is the foundation for many continuous-time stochastic processes and is widely used in financial mathematics and physics

Applications and Examples

• Queueing theory: Markov chains and Poisson processes are used to model and analyze queueing systems (e.g., customer service centers, manufacturing systems)
  • Example: M/M/1 queue with Poisson arrivals and exponentially distributed service times
• Reliability theory: Stochastic processes are used to model the lifetime and failure behavior of systems and components
  • Example: exponential distribution for modeling the time to failure of a light bulb
• Finance: Stochastic processes, particularly Brownian motion, are used to model asset prices, interest rates, and other financial variables
  • Example: Black-Scholes model for pricing European options using geometric Brownian motion
• Biology: Stochastic processes are used to model population dynamics, genetic drift, and the spread of epidemics
  • Example: birth-death processes for modeling population growth and extinction
• Physics: Brownian motion is used to describe the random motion of particles suspended in a fluid
  • Example: diffusion of molecules in a gas or liquid
• Speech recognition: Hidden Markov models (HMMs) are used to model and recognize speech patterns
  • Example: using HMMs to identify phonemes and words in a spoken sentence

Problem-Solving Techniques

• Identify the type of stochastic process based on the problem description and the properties of the process
• Determine the state space (discrete or continuous) and the index set (discrete or continuous time)
• For Markov chains:
  • Construct the transition probability matrix $P$
  • Use the Chapman-Kolmogorov equations to find $n$-step transition probabilities
  • Solve for the stationary distribution $\pi$ by setting up and solving the system of linear equations $\pi P = \pi$ and $\sum_{i \in S} \pi_i = 1$
• For Poisson processes:
  • Identify the rate parameter $\lambda$
  • Use the Poisson distribution to find probabilities of the number of events in a given time interval
  • Use the exponential distribution to find probabilities related to inter-arrival times
• For Brownian motion:
  • Use the properties of normal distribution to find probabilities related to increments
  • Apply Itô's lemma for transformations of Brownian motion (e.g., geometric Brownian motion)
• Simulate sample paths of stochastic processes using appropriate methods (e.g., inverse transform method for Poisson processes, Euler-Maruyama scheme for Brownian motion)
• Use conditioning and the law of total probability to break down complex problems into simpler subproblems
• Apply moment-generating functions or characteristic functions to derive properties of stochastic processes