# Levy exponential local martingale

We will review a sufficient condition for Levy stochastic exponential to be a local martingale.

Let Levy-type stochastic integral ${Y}$ be given by

$\displaystyle d Y(t) = G(t) dt + F(t) dW(t) + H(t,x) \tilde N(dt, dx) + K(t,x) N(dt, dx). \ \ \ \ \ (1)$

Our first goal is to find necessary and sufficient conditions for a Levy-type stochastic integral ${Y}$ to be a martingale. First, we impose some conditions on ${K}$ and ${G}$ with the notion of ${\mathcal{P}_2(T,E)}$ and ${\mathcal{H}_2(T)}$ (see definitions here).

1. (LM1) ${K\in \mathcal{P}_2(t, B_1^c(0))}$ for each ${t>0}$.
2. (LM2) ${\mathbb{E} [\int_0^t \int_{|x|\ge 1} |K(s,x)| \nu(dx) ds ] <\infty}$.
3. (LM3) ${G^{1/2} \in \mathcal{H}_2(t)}$ for each ${t>0}$.

It follows from (LM2) that ${\int_0^t \int_{|x|\ge 1} |K(s,x)| \nu(dx) ds <\infty}$ a.s. and we can write

$\displaystyle \int_0^t \int_{|x|\ge 1} K(s,x) N(ds, dx) = \int_0^t \int_{|x|\ge 1} K(s,x) \tilde N(ds, dx) + \int_0^t \int_{|x|\ge 1} K(s,x) \nu(dx) ds,$

for each ${t\ge 0}$ with the compensated integral being a local martingale.

Theorem 1 If ${Y}$ is a Levy-type stochastic integral of the form (1) and the assumptions (LM1)–(LM3) are satisfied, then ${Y}$ is a local martingale if and only if

$\displaystyle G(t) + \int_{|x|\ge 1} K(t,x) \nu(dx) = 0 \hbox{ a.s.}$

for Lebesgue almost all ${t\ge 0}$.

Directly applying Ito’s formula on ${e^{Y(t)}}$, we have

Corollary 2 (Exponential martingale) With the same condition as Theorem 1, ${e^Y}$ is a local martingale if and only if

$\displaystyle G(s) + \frac 1 2 F^2(s) + \int_{|x|<1} (e^{H(s,x)} -1 - H(s,x)) \nu(dx) + \int_{|x|\ge 1} (e^{K(s,x)} -1) \nu(dx) = 0,$

almost surely and for almost all ${s\ge 0}$.

So, ${e^Y}$ is a local martingale if and only if for ${t\ge 0}$,

$\displaystyle \begin{array}{ll} e^{Y(t)} & = 1 + \int_0^t e^{Y(s-)} F(s) d W(s) + \int_0^t \int_{|x|<1} e^{Y(s-)} (e^{H(s,x)} -1) \tilde N(ds, dx) \\ & \hspace{2in} + \int_0^t \int_{|x|\ge 1} e^{Y(s-)} (e^{K(s,x)} -1) \tilde N(ds, dx). \end{array} \ \ \ \ \ (2)$

The process ${e^Y}$ given by equation (2) is called an exponential local martingale. Two important examples are:

1. The Brownian case: If ${Y}$ is a Brownian integral of

$\displaystyle Y(t) = \int_0^t F(s) dW(s) + \int_0^t G(s) ds,$

where ${G(t) = - \frac 1 2 F^2(s)}$. Then, ${e^Y}$ is exponential local martingale. A sufficient condition to be a martingale is the well-known Novikov condiion, i.e.

$\displaystyle \mathbb{E} \Big[ \exp\Big\{\int_0^\infty \frac 1 2 F^2(s) ds \Big\}\Big] <\infty.$

2. The Poisson case: If ${Y}$ is a Poisson integral driven by a Poisson process ${N}$ of intensity ${\lambda}$ and has the form of

$\displaystyle Y(t) = \int_0^t K(s) d N(s) + \int_0^t G(s) ds,$

If ${G(t) = -\lambda (e^{K(t)} -1) }$ almost surely in ${t}$, then we have ${e^Y}$ as exponential local martingale.

A general exponential local martingale of the form ${\exp \{M(t) - \frac 1 2 \langle M, M\rangle(t)\}, t\ge 0)}$ to be a martingale, some additional conditions are needed, see [RY99], pp307–309. In particualr, One may expect exponential martingale in Theorem 1 under following (M1)-(M2) in place of (LM1)-(LM3),

1. (M1) ${\mathbb{E} [\int_0^t \int_{|x|\ge 1} \max \{ |K(s,x)|, |K(s,x)|^2 \} \nu(dx) ds ] <\infty}$.
2. (M2) ${\int_0^t \mathbb{E} [|G(s)| ] ds < \infty}$ for each ${t>0}$.