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

Let Levy-type stochastic integral be given by

Our first goal is to find necessary and sufficient conditions for a Levy-type stochastic integral to be a martingale. First, we impose some conditions on and with the notion of and (see definitions here).

- (LM1) for each .
- (LM2) .
- (LM3) for each .

It follows from (LM2) that a.s. and we can write

for each with the compensated integral being a local martingale.

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

for Lebesgue almost all .

Directly applying Ito’s formula on , we have

Corollary 2 (Exponential martingale)With the same condition as Theorem 1, is a local martingale if and only if

almost surely and for almost all .

So, is a local martingale if and only if for ,

The process given by equation (2) is called an *exponential local martingale.* Two important examples are:

- The Brownian case: If is a Brownian integral of
where . Then, is exponential local martingale. A sufficient condition to be a martingale is the well-known

*Novikov condiion*, i.e. - The Poisson case: If is a Poisson integral driven by a Poisson process of intensity and has the form of
If almost surely in , then we have as exponential local martingale.

A general exponential local martingale of the form 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),

- (M1) .
- (M2) for each .