Now that it has been shown that stochastic integration can be performed with respect to any local martingale, we can move on to the following important result. *Stochastic integration preserves the local martingale property*. At least, this is true under very mild hypotheses. That the martingale property is preserved under integration of bounded elementary processes is straightforward. The generalization to predictable integrands can be achieved using a limiting argument. It is necessary, however, to restrict to locally bounded integrands and, for the sake of generality, I start with local sub and supermartingales.

Theorem 1LetXbe a local submartingale (resp., local supermartingale) and be a nonnegative and locally bounded predictable process. Then, is a local submartingale (resp., local supermartingale).

*Proof:* We only need to consider the case where *X* is a local submartingale, as the result will also follow for supermartingales by applying to *-X*. By localization, we may suppose that is uniformly bounded and that *X* is a proper submartingale. So, for some constant *K*. Then, as previously shown there exists a sequence of elementary predictable processes such that converges to in the semimartingale topology and, hence, converges ucp. We may replace by if necessary so that, being nonnegative elementary integrals of a submartingale, will be submartingales. Also, . Recall that a cadlag adapted process *X* is locally integrable if and only its jump process is locally integrable, and all local submartingales are locally integrable. So,

is locally integrable. Then, by ucp convergence for local submartingales, *Y* will satisfy the local submartingale property. ⬜

For local martingales, applying this result to gives,

Theorem 2LetXbe a local martingale and be a locally bounded predictable process. Then, is a local martingale.

This result can immediately be extended to the class of local -integrable martingales, denoted by .

Corollary 3Let for some and be a locally bounded predictable process. Then, .

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