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