Lévy Processes and Jump Diffusions

Financial markets crash in an instant-but Brownian motion cannot "jump". Lévy processes fix this: they include jumps of any size, describing market crashes, earthquakes, and internet traffic in a single mathematical framework.

• **Finance:** jump-diffusion models (Merton jump-diffusion, Variance Gamma) - option pricing under tail risk
• **Insurance:** compound Poisson process in the Cramér - Lundberg model for ruin probability estimation
• **Physics:** anomalous diffusion in heterogeneous media described by α-stable processes with index α < 2

Предварительные знания

• Poisson Process
• Brownian Motion

Definition and Properties of Lévy Processes

Jump-diffusion models for stock prices use Lévy processes: Merton's model (1976) adds Poisson jumps to Brownian motion. Bloomberg terminals price exotic options via Lévy process simulation at 1M+ contracts daily. The variance-gamma process, a popular Lévy model, fits S&P 500 returns with 3 parameters.

1. X(0) = 0 a.s. 2. Independent increments: X(t) - X(s) ⊥ X(s) - X(r) for r < s < t 3. Stationary increments: X(t+s) - X(s) ~ X(t) 4. Stochastic continuity: X(t+ε) → X(t) in probability

Key property: the distribution of X(t) is **infinitely divisible**: for any n, there exist i.i.d. X_1,...,X_n such that X(t) ~ X_1 + ... + X_n.

The Lévy - Khintchine Formula

The characteristic function of any Lévy process is completely described by the **Lévy - Khintchine formula**. This is the "passport" of infinitely divisible distributions.

A Lévy process = Brownian component (σ²) + small jumps (∫_{|x|<1}) + large jumps (∫_{|x|≥1}) + linear drift (b). The Lévy - Itô decomposition makes this precise.

For the Poisson process: ν = λ·δ_1, σ = 0. For Brownian motion: ν = 0, σ² > 0.

Compound Poisson Process

The **compound Poisson process** is the simplest Lévy process with variable-size jumps. It is widely used in insurance (Cramér - Lundberg model) and finance.

Insurer's surplus: R(t) = u + ct - X(t), where u is initial capital, c is premium rate, X(t) is a compound Poisson process. The ruin probability ψ(u) = P(R(t) < 0 for some t) decreases exponentially as u → ∞.

Stable Distributions and α-Stable Processes

**Stable distributions** generalize the normal law: sums of i.i.d. variables (after normalization) converge to a stable distribution. They describe heavy tails in finance, physics, and networks.

If X_i are i.i.d. with tails P(|X| > x) ~ L(x)·x^{-α}, then (X_1+...+X_n)/n^{1/α} → X_α. The classical CLT is the case α=2. For α < 2 the variance is infinite, and n^{1/2} normalization fails.

Key Ideas

• **Lévy process**: independent stationary increments, càdlàg; generalizes BM and the Poisson process
• **Lévy - Khintchine formula**: triplet (b, σ², ν) fully characterizes the process via its characteristic function
• **Compound Poisson**: random-size jumps; E[X(t)] = tλE[Y], Var = tλE[Y²]
• **α-stable distributions**: heavy tails, generalized CLT; variance finite only at α = 2

Related Topics

Lévy processes unify several key areas of stochastic calculus:

• Poisson Process — Special case of a Lévy process (Lévy measure is a point mass)
• Brownian Motion — Continuous Lévy process, the Gaussian component
• Itô Stochastic Integrals — Integration with respect to Lévy processes extends Itô's theory

Вопросы для размышления

• Why does the Lévy - Khintchine formula include the compensating term iux·1_{|x|<1}? What would happen without it?
• How would one estimate the parameters of a compound Poisson process from insurance claims data?
• When α = 1.5 the variance is infinite - how does this practically affect financial risk models?

Связанные уроки

• prob-10-continuous