Advanced Theory

6-10 Hours

Independent Increment Processes

Lévy processes and the general theory of processes with independent increments

Definition & Properties - Independent Increments

A process $\{X(t), t \geq 0\}$ has independent increments if for any $0 \leq t_1 < t_2 < \dots < t_n$ , the random variables:

X(t_1) - X(t_0), X(t_2) - X(t_1), \dots, X(t_n) - X(t_{n-1})

are mutually independent (where $t_0 = 0, X(0) = 0$ usually).

Key Properties:

All independent increment processes are Markov processes
Mean function is additive: $\mu_X(t+s) = \mu_X(t) + \mu_X(s)$
Covariance simplifies: $C_X(t_1, t_2) = \text{Var}(X(\min(t_1, t_2)))$

Stationary Independent Increments

If the distribution of $X(t+h) - X(t)$ depends only on $h$ (not on $t$ ), the process has stationary independent increments (Lévy processes).

Examples: Poisson processes, Brownian motion, compound Poisson processes.

Lévy Processes - Definition

A Lévy process is a continuous-time stochastic process $\{X(t), t \geq 0\}$ with:

Stationary independent increments
$X(0) = 0$ (starts at zero)
Stochastic continuity: $\lim_{h \to 0} P(|X(t+h) - X(t)| > \epsilon) = 0$

Examples: Brownian motion, Poisson processes, compound Poisson processes, Gamma processes.

Compound Poisson Process - Definition

A compound Poisson process is defined as:

X(t) = \sum_{i=1}^{N(t)} Y_i

where $N(t)$ is a Poisson process with rate $\lambda$ and $Y_i$ are i.i.d. random variables (jump sizes).

Applications: Insurance claims (random claim amounts), portfolio returns (random trade profits/losses).

Fundamental Theorems - Lévy-Khintchine Representation

The characteristic function of a Lévy process $X(t)$ is:

E[e^{iuX(t)}] = e^{t\psi(u)}

where $\psi(u)$ is the Lévy exponent, completely characterizing the process through drift, diffusion, and jump components.

Worked Examples - Example 1: Simple Random Walk

Problem:

Let $X_n = \sum_{i=1}^n Y_i$ where $Y_i$ are i.i.d. Show this is an independent increment process.

Solution:

Increments $X_{n+1} - X_n = Y_{n+1}$ are independent (since $Y_i$ are i.i.d.), so the process has independent increments.

Example 2: Compound Poisson Process Mean

Problem:

For $X(t) = \sum_{i=1}^{N(t)} Y_i$ where $N(t)$ is Poisson( $\lambda t$ ) and $E[Y_i] = \mu$ , find $E[X(t)]$ .

Solution:

Using conditional expectation: $E[X(t)] = E[E[X(t)|N(t)]] = E[N(t) \mu] = \lambda t \mu$ .

Example 3: Covariance of Independent Increment Process

Problem:

For an independent increment process with $X(0) = 0$ , find $\text{Cov}(X(t_1), X(t_2))$ for $t_1 < t_2$ .

Solution:

Write $X(t_2) = X(t_1) + [X(t_2) - X(t_1)]$ . Using independent increments: $\text{Cov}(X(t_1), X(t_2)) = \text{Var}(X(t_1))$ .

Generally: $C_X(t_1, t_2) = \text{Var}(X(\min(t_1, t_2)))$ .

Practice Quiz

Questions

Correct

Accuracy

What is an independent increment process?

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What is a Lévy process?

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Are all independent increment processes also Markov processes?

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What is a compound Poisson process?

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What does 'infinitely divisible' mean for a distribution?

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What is the difference between independent increments and stationary increments?

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What is a Gamma process?

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What is the Lévy-Khintchine representation?

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What makes a process 'self-similar'?

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What is the covariance structure of an independent increment process?

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Frequently Asked Questions

What is the difference between independent increments and stationary increments?

Independent increments means the changes over non-overlapping time intervals are independent random variables. Stationary increments means the distribution of X(t+h)-X(t) depends only on h, not on t. A process can have one property without the other, though Poisson processes have both.

Are all independent increment processes also Markov processes?

Yes! Independent increments is a stronger condition than the Markov property. If future increments are independent of the past, then the future is independent of the past given the present, which is exactly the Markov property. However, not all Markov processes have independent increments.

What is a Lévy process?

A Lévy process is a continuous-time stochastic process with stationary independent increments that starts at zero and is stochastically continuous. It's the continuous-time analog of a random walk. Brownian motion and Poisson processes are both Lévy processes.

What does 'infinitely divisible' mean?

A distribution is infinitely divisible if for any n, it can be represented as the sum of n i.i.d. random variables. The distributions of increments in Lévy processes are always infinitely divisible. This connects to the Lévy-Khintchine representation.

How does a compound Poisson process differ from a regular Poisson process?

A regular Poisson process counts events, with each event adding +1. A compound Poisson process adds random 'jump sizes' Yᵢ at each event time. So X(t) = Σᵢ₌₁^{N(t)} Yᵢ, where N(t) is Poisson and Yᵢ are i.i.d. This models scenarios like insurance claims (random claim amounts) or portfolio returns (random trade profits/losses).