Independent Increment Processes Formulas

Definition

A stochastic process $\{X(t); t \geq 0\}$ with state space $\mathbb{R}$ or $\mathbb{Z}$ is an independent increment process if for any positive integer $n$ and any time points $0 = t_0 < t_1 < t_2 < \cdots < t_n$, the increments are independent:

Key Assumption: Often $X(0) = 0$ for simplicity

Markov Property

Independent increment processes are necessarily Markov processes:

Intuition: Future depends only on present, not on past

Mean Function Additivity

For any $t > s \geq 0$:

Derivation: $X(t) = X(s) + [X(t) - X(s)]$ and linearity of expectation

Increment Expectation

Rearranging the additivity property:

Application: Calculate expected increments from process values

Mean Square Value

Relationship between mean square value and variance:

Note: $D_X(t) = \text{Var}[X(t)]$ is the variance function

Covariance Simplification

For any $s, t \geq 0$:

Key Result: Covariance depends only on the minimum time

Derivation

For $s < t$, $X(t) = X(s) + [X(t) - X(s)]$:

Reasoning: Increment is independent of $X(s)$

Autocorrelation Function

The autocorrelation function is:

Property: Correlation depends on the ratio of variances

Simple Random Walk Definition

Let $X_1, X_2, \ldots$ be i.i.d. with $P(X_i = 1) = p$ and $P(X_i = -1) = q = 1 - p$:

Process: $\{S_n; n = 1, 2, \ldots\}$ is a discrete-time independent increment process

Mean Function

Expected value after $n$ steps:

Special Case: Fair game ($p = q = 0.5$) gives $E[S_n] = 0$

Variance Function

Variance after $n$ steps:

Special Case: Fair game gives $D[S_n] = n$

Covariance Function

For $n \leq m$:

Result: $\text{Cov}[S_n, S_m] = D[S_n]$ by covariance simplification

Increment Probability

For independent increments:

Application: Calculate joint probabilities using independence

Normal Distribution Properties

If $X(t) \sim N(\mu_t, \sigma_t^2)$ and $X(s) \sim N(\mu_s, \sigma_s^2)$ with independent increments:

Note: Variances add for independent normal variables

Stationary Increments

If increments are also stationary (distribution depends only on time difference):

Implication: Process has time-homogeneous properties

Characteristic Function

For independent increments, the characteristic function factors:

Application: Analyze distribution properties through characteristic functions

Poisson Process

A counting process $\{N(t); t \geq 0\}$ with:

• Independent increments
• Stationary increments
• $N(t) - N(s) \sim \text{Poisson}(\lambda(t - s))$

Property: $E[N(t)] = \lambda t$ and $D[N(t)] = \lambda t$

Brownian Motion

A continuous process $\{B(t); t \geq 0\}$ with:

• Independent increments
• Stationary increments
• $B(t) - B(s) \sim N(0, \sigma^2(t - s))$

Property: $E[B(t)] = 0$ and $D[B(t)] = \sigma^2 t$

General Form

For independent increment processes with $X(0) = 0$:

Decomposition: Process can be decomposed into independent increments