Basic definitions

We will review definition and some basic properties of Levy process.

The characteristic function of r.v. {X\sim \mu} is the mapping {\Phi: \mathbb{R}^d \rightarrow \mathbb{C}} defined by

\displaystyle  \Phi(u) = \mathbb{E} (e^{iu\cdot X}) = \int_{\mathbb{R}^d} e^{iu\cdot y} \mu(dy).

A random variable {X} is infinitely divisible if its law {p_x} is infinitely divisible, e.g. {X =^d Y_1^{(n)}+\cdots + Y_n^{(n)}}, where {Y_1^{(n)}, \ldots Y_n^{(n)}} are i.i.d., for each {n\in \mathbb{N}}. Note that, the characteristic function of {X} can be written as {\Phi_X(u) = (\Phi_{Y_1^{(n)}}(u))^n} in the above.

Definition 1 Levy measure is a measure {\nu} on {\mathbb{R}^d \setminus \{0\}} such that

\displaystyle   \int (|y|^2 \wedge 1) \nu (dy) < \infty. \ \ \ \ \ (1)

The characteristic functions of infinitely divisible probability measures were completely characterized by Levy and Khintchine in the 1930s. This is a fundamental result.

Theorem 2 (Levy-Khintchine) A Borel probability measure {\mu} on {\mathbb{R}^d} is infinitely divisible if there exists a vector {b\in \mathbb{R}^d}, a non-negative symmetric {d\times d} matrix {A} and a Levy measure {\nu} on {\mathbb{R}^d\setminus \{0\}} such that for all {u\in \mathbb{R}^d},

\displaystyle  \phi_\nu(u) = \exp \Big\{i b\cdot u - \frac 1 2 u \cdot A u + \int_{\mathbb{R}^d \setminus \{0\}} \Big(e^{iu\cdot y} - 1 - iu\cdot y 1_{B_1(0)}(y)\Big) \nu (dy) \Big\}. \ \ \ \ \ (2)

Conversely, any mapping of the form (2) is the characteristic function of an infinitely divisible probability measure on {\mathbb{R}^d}.

The triple {(b, A, \nu)} is called the characteristics of the infinitely divisible random variable {X}, and {\eta := \log \phi_{\mu}} is called the Levy symbol or Characteristic exponent.

Definition 3 A Levy process {X = (X(t), t\ge 0)} is a stochastic process satisfying the following:

  1. {X(0) = 0} w.p.1
  2. {X} has independent and stationary increments,
  3. {X} is stochastically continuous, i.e. for all {a>0} and for all {s\ge 0}, {\lim_{t\rightarrow s} P(|X(t) - X(s)|>a) = 0}.

It follows from the Definition 3 (2) that each Levy process {X(t)} is infinitely divisible. Therefore,

Theorem 4 If {X} is a Levy process, then {X(t)} is infinitely divisible for each {t\ge 0}. Furthermore,

\displaystyle  \phi_{X(t)} (u) = e^{t\eta(u)}, \forall u\in \mathbb{R}^d, t\ge 0,

where {\eta} is the Levy symbol of {X(1)}.

Here are examples of Levy processes.

Example 1 (Brownian motion with drift) Let {B(t)} be a standard Brownian motion in {\mathbb{R}^d}. Then, {C(t) = bt + \sigma B(t)} is a Levy process with its characteristic function given by

\displaystyle \Phi_{B(t)} (u) = \exp \{ t \eta_C(u)\},

where {\eta_C(u)} is Levy symbol of {C(1)} of the form

\displaystyle \eta_C(u) = i(b,u) - \frac 1 2 (u, \sigma \sigma^T u).

In fact a Levy process has a continuous sample paths if and only if it is of the form {C(t)}.

Example 2 (The compounded Poisson process) Let {\{Z(n): n\in \mathbb{N}\}} be a sequence of i.i.d r.v. in {\mathbb{R}^d} with law {\mu_Z}. Let {N} be a Poisson process of intensity {\lambda}, which is independent of {Z(n)}. The compound Poisson process {Y} is defined as follows:

\displaystyle Y(t) = Z(1) + \cdots + Z(N(t)).

Then, {Y} has Levy symbol of

\displaystyle \eta_Y(u) = \int (e^{i(u,y)} - 1) \lambda \mu_Z(dy).

If {\mu_Z = \delta_{1}}, then {Y} is said to be Poisson process. The characteristics of {Y(1)} is

\displaystyle b = \lambda \mathbb E[Z(1) I_{B_{1}(0)}(Z(1))], \ A = 0, \ \nu = \lambda \mu_{Z}.

Example 3 (Subordinators) A subordinator is a one-dimensional Levy process which is increasing a.s. Such processes can be thought of as a random model of time eveolution. A Levy process {T = \{T(t): t\ge 0\}} be a subordinator if and only if its Levy symbol takes the form

\displaystyle \eta (u) = i b u + \int_{(0,\infty)} (e^{iuy} - 1) \lambda(dy),

for some {b\ge 0} and a Levy measure {\lambda} satisfying

\displaystyle \lambda(-\infty, 0) = 0, \quad \int_{(0,\infty)} (y\wedge 1) \lambda(dy) <\infty.

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2 comments

  1. Pingback: Levy Process-2: Levy-Ito decomposion « 01law's Blog


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