A stochastic process is a semimartingale if and only if it can be decomposed as the sum of a local martingale and an FV process. This is stated by the Bichteler-Dellacherie theorem or, alternatively, is often taken as the definition of a semimartingale. For *continuous* semimartingales, which are the subject of this post, things simplify considerably. The terms in the decomposition can be taken to be continuous, in which case they are also unique. As usual, we work with respect to a complete filtered probability space , all processes are real-valued, and two processes are considered to be the same if they are indistinguishable.

Theorem 1A continuous stochastic processXis a semimartingale if and only if it decomposes as

(1)

for a continuous local martingaleMand continuous FV processA. Furthermore, assuming that, decomposition (1) is unique.

*Proof:* As sums of local martingales and FV processes are semimartingales, *X* is a semimartingale whenever it satisfies the decomposition (1). Furthermore, if were two such decompositions with then is both a local martingale and a continuous FV process. Therefore, is constant, so and .

It just remains to prove the existence of decomposition (1). However, *X* is continuous and, hence, is locally square integrable. So, Lemmas 4 and 5 of the previous post say that we can decompose where *M* is a local martingale, *A* is an FV process and the quadratic covariation is a local martingale. As *X* is continuous we have so that, by the properties of covariations,

(2) |

We have shown that is a nonnegative local martingale so, in particular, it is a supermartingale. This gives . Then (2) implies that is zero and, hence, *A* and are continuous. ⬜

Using decomposition (1), it can be shown that a predictable process is *X*-integrable *if and only if* it is both *M*-integrable and *A*-integrable. Then, the integral with respect to *X* breaks down into the sum of the integrals with respect to *M* and *A*. This greatly simplifies the construction of the stochastic integral for continuous semimartingales. The integral with respect to the continuous FV process *A* is equivalent to Lebesgue-Stieltjes integration along sample paths, and it is possible to construct the integral with respect to the continuous local martingale *M* for the full set of *M*-integrable integrands using the Ito isometry. Many introductions to stochastic calculus focus on integration with respect to continuous semimartingales, which is made much easier because of these results.

Theorem 2Letbe the decomposition of the continuous semimartingaleXinto a continuous local martingaleMand continuous FV processA. Then, a predictable processisX-integrable if and only if

(3)

almost surely, for each time. In that case,is bothM-integrable andA-integrable and,

(4)

gives the decomposition ofinto its local martingale and FV terms.