A Lévy process is a stochastic process that extends Brownian motion by allowing both continuous paths and discontinuous jumps. It is rigorously defined as a real-valued, adapted stochastic process \( L = \{ L_t, t \geq 0 \} \) on a probability space \( (\Omega, \mathcal{F}, P) \) that satisfies the following properties:
- Independent increments: The increments of the process are independent, meaning that for any \( 0 \leq s < t \leq T \), the increment \( L_t - L_s \) is independent of the past information up to time \( s \).
- Stationary increments: The distribution of an increment \( L_{t+s} - L_t \) depends only on the time gap \( s \), and not on the absolute value of \( t \).
- Stochastic continuity: The probability of a large variation over an infinitesimally small interval diminishes as the interval approaches zero. Formally, for any \( \epsilon > 0 \) \[ \lim_{s \to t} P(|L_t - L_s| > \epsilon) = 0. \]
These three fundamental properties ensure that Lévy processes provide a flexible framework for modeling real-world stochastic phenomena, particularly in finance, where asset prices can exhibit both continuous fluctuations and sudden jumps.
The Characteristic Function of a Lévy Process
A key feature of Lévy processes is the Lévy-Khintchine representation. The characteristic function of a process \( L_t \) is defined as:
\[ \phi(u) = \mathbb{E}[e^{iu L_t}] \]
It represents the Fourier transform of the probability density and encapsulates all the information about the process’s distribution. For a process with independent and stationary increments, it takes an exponential form:
\[ \phi_t(u) = e^{t \psi(u)} \]
where \( \psi(u) \) is the characteristic exponent.
The Case of Brownian Motion
For standard Brownian motion \( B_t \sim \mathcal{N}(0, t) \), the characteristic function is given by:
\[ \mathbb{E}[e^{iu B_t}] = e^{- \frac{1}{2} u^2 t} \]
The characteristic exponent is:
\[ \psi(u) = - \frac{1}{2} u^2 \]
Adding a Drift
By adding a drift \( \mu \), the process becomes:
\[ L_t = \mu t + \sigma B_t \]
This modifies the characteristic function to:
\[ \mathbb{E}[e^{iu L_t}] = e^{iu \mu t} e^{- \frac{1}{2} \sigma^2 u^2 t} \]
Introducing Jumps
To incorporate jumps, we introduce a Lévy measure \( \nu(dx) \), which describes the intensity and distribution of jumps. This modifies the characteristic exponent to:
\[ \psi(u) = i \mu u - \frac{1}{2} \sigma^2 u^2 + \int (e^{iux} - 1 - iux 1_{|x|\leq1}) \nu(dx) \]
The Lévy Condition and Its Significance
The Lévy condition ensures that the integral in the Lévy-Khintchine representation remains well-defined. The condition is expressed as:
\[ \int (1 \wedge x^2) \nu(dx) < \infty \]
The term \( 1 \wedge x^2 \) signifies that for small jumps (when \( x \to 0 \)), the integral is dominated by \( x^2 \), preventing divergence. For large jumps, the condition simply ensures that the integral remains finite. This balance allows Lévy processes to model both continuous paths and discontinuous jumps in a mathematically rigorous manner.
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