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 1 A continuous stochastic process X is a semimartingale if and only if it decomposes as
for a continuous local martingale M and continuous FV process A. 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,
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 2 Let be the decomposition of the continuous semimartingale X into a continuous local martingale M and continuous FV process A. Then, a predictable process is X-integrable if and only if
almost surely, for each time . In that case, is both M-integrable and A-integrable and,
gives the decomposition of into its local martingale and FV terms.
Proof: First, suppose that (3) holds. As is finite, is M-integrable. Also, as is finite, is A-integrable and agrees with the Lebesgue-Stieltjes integral. As is integrable with respect to both M and A, it is integrable with respect to X. By preservation of the local martingale property, the term in decomposition (4) is a continuous local martingale. Also, as it agrees with the Lebesgue-Stieltjes integral along sample paths, is an FV process.
It still needs to be shown that inequality (3) holds whenever is X-integrable. As A is a continuous FV process, it does not contribute to quadratic variations, so . Commuting stochastic integration and quadratic variations,
So, the first term on the left of (3) is finite.
As is a continuous semimartingale, it decomposes as for a continuous local martingale N and FV process B, which can be assumed to start from zero. Setting , the fact that and are bounded predictable processes gives
Uniqueness of the decomposition into continuous local martingale and FV terms implies that and are equivalent. Let us denote this process by C. Then, applying associativity of integration to the Lebesgue-Stieltjes integrals,
So, the second term on the left of (3) is also finite. ⬜
A particular consequence of Theorem 2 is that, for a continuous FV process A and predictable , then is A-integrable if and only if it is A-integrable in the Lebesgue-Stieltjes sense. That is, if is finite. Then, the stochastic integral agrees with the Lebesgue-Stieltjes integral along the sample paths. So, for continuous FV processes, stochastic integration does not improve upon standard, non-deterministic, Lebesgue-Stieltjes integration. This might not sound very surprising, but it is not true for general FV processes. There exist non-continuous FV processes A and predictable which are A-integrable in the stochastic sense, but is not well-defined as a Lebesgue-Stieltjes integral (see Failure of Pathwise Integration for FV Processes). It is also possible to give a more direct proof that stochastic integration coincides with Lebesgue-Stieltjes integration for continuous FV processes without relying on (1). This can be done by applying the Jordan decomposition, as I will show below.
Theorem 1 also has the interesting consequence that continuous FV processes are the only semimartingales with zero quadratic variation.
Lemma 3 Let X be a stochastic process. Then, the following are equivalent.
- X is a continuous FV process.
- X is a semimartingale with zero quadratic variation.
- X is a continuous semimartingale such that for all (continuous) local martingales M.
Proof: It was previously shown that all continuous FV processes are semimartingales with zero quadratic variation. Next, if X is a semimartingale with zero quadratic variation then , so X is continuous. Also, the Cauchy-Schwarz inequality gives
for all local martingales M. So, it only needs to be shown that, if X is a continuous semimartingale with zero quadratic covariation against all continuous local martingales then it is an FV process. Let be decomposition (1). As A is an FV process, the quadratic covariation is zero. Also, by hypothesis, is zero. So, M is a continuous local martingale with quadratic variation and, hence is constant. Therefore, is an FV process. ⬜
It is also possible to describe integration with respect to continuous FV processes in terms of their Jordan decomposition applied to the sample paths. If A is an FV process, this allows us to write for increasing processes and starting from zero. This can be done in such a way that and are minimal, in which case I refer to them as the increasing and decreasing parts of A respectively. Alternatively, and are the minimum nonnegative processes such that and are increasing. The variation of A up until a time t is given by . As A is right-continuous, are also right-continuous. It can also be seen that they are measurable and adapted by computing them along a partition. Choosing times then,
The limits here are all taken as and as the mesh of the partition, , goes to zero. So, are adapted increasing and right-continuous processes. The Hahn decomposition theorem implies that there exists a measurable process with such that , and, consequently, . In general, will not be predictable. However, in the case where A is a continuous FV process then will be continuous and the following lemma shows that can be taken to be predictable.
Lemma 4 Let A be a continuous FV process. Then, there exists a predictable process with such that is increasing.
Furthermore, if A is any FV process and is measurable such that and is increasing, then it follows that , and are respectively the variation, increasing part and decreasing parts of A.
Proof: We start by showing that dA is absolutely continuous with respect to dV. That is, if is a nonnegative bounded predictable process such that then we need to show that is zero. As and are increasing, this gives
so that as required. As previously shown, this means that for some predictable process which is V-integrable in the Lebesgue-Stieltjes sense. So, setting , we have
which is increasing.
Now suppose that A is an FV process, and that is increasing. Integrating and with respect to shows that the processes and are increasing. If was any other increasing process starting from zero such that is increasing then we would have
implying that as claimed. Applying the same result to shows that and, therefore, . ⬜
We now give a quick proof that stochastic integration and pathwise Lebesgue-Stieltjes integration coincides for a large class of FV processes, without relying on decomposition (1). The statement of Lemma 5 trivially applies to all increasing FV processes and, applying the lemma above, it also applies to all continuous FV processes.
Lemma 5 Let A be an FV process such that is increasing for some bounded predictable and nowhere-zero process .
Then, a predictable process is A-integrable if and only if (almost surely) for each time . Furthermore, the stochastic and Lebesgue-Stieltjes integrals coincide with probability one.
Proof: Replacing by if necessary, we may suppose that . Then, Lemma 4 says that is equal to the variation of A. For any A-integrable process (in the sense of stochastic integration), choose a sequence of nonnegative bounded predictable processes increasing to . We have
The first equality here is just monotone convergence for Lebesgue-Stieltjes integration, and the second equality is substituting for the variation of A. However, as is bounded, the integral is well-defined and identical in the Lebesgue-Stieltjes and stochastic senses. The last equality is dominated convergence for the stochastic integral and, since we know that is A-integrable it follows that is finite. So, is A-integrable in the Lebesgue-Stieltjes sense, and the stochastic and Lebesgue-Stieltjes integrals coincide. ⬜
The decomposition of a continuous semimartingale into its local martingale and FV components is quite robust and, in particular, is continuous in the semimartingale topology.
Lemma 6 Let , X be continuous local martingales and , be their decompositions (1) into continuous local martingale and continuous FV terms with . Then, the following are equivalent as n goes to infinity.
- in the semimartingale topology.
- and in the semimartingale topology.
- The following limit holds in probability for each
Proof: Without loss of generality we can assume that by replacing with respectively, if necessary.
Starting with the third condition, by lemma 6 of the post on continuous local martingales, this implies that in the semimartingale topology. Furthermore, for any sequence of elementary predictable processes bounded by 1, is bounded by the variation of A up to time t, which tends to zero in probability. So, by definition, in the semimartingale topology, giving the second condition.
That the second condition implies the first is clear from the definition of semimartingale convergence, which is actually a vector topology.
Finally, suppose that the first condition holds. As convergence in the semimartingale topology in particular implies convergence in probability at each time, we see that tends to zero in probability By the continuity of quadratic variations in the semimartingale topology, tends to zero in probability. So, again applying lemma 6 of the post on continuous local martingales, this shows that, under the semimartingale topology, and, hence, . By lemma 4, there exists a sequence of predictable processes bounded by 1 such that is equal to the variation of A. Then, by semimartingale convergence, it follows that tends to zero in probability, giving the third statement. ⬜
We previously saw that stochastic integration satisfies dominated convergence, where the integrals of a dominated sequence of integrals converges in the semimartingale topology. For continuous semimartingales, a weaker condition than dominated convergence can be used, and we obtain an ‘if and only if’ condition for the integral to converge.
Theorem 7 Let X be a continuous local martingale and be decomposition (1). Then, if and are X-integrable processes.
Then, in the semimartingale topology if and only if
in probability for each t.
shows that in the semimartingale topology if and only if (5) tends to zero in probability. ⬜
Stochastic integration, as originally developed by Kiyoshi Ito, gave a rigorous construction of the integral for a predictable process and Brownian motion B. Combining this with the standard Lebesgue integral with respect to time leads us to consider processes of the form
Here, B is a Brownian motion defined on the underlying filtered probability space and , are predictable processes. For this expression to be well defined, we require and to be almost surely finite for all times t. Processes of the form (6) are known as Ito processes.
According to this definition, being constructed as integrals with respect to Brownian motion, Ito processes can appear to be a very specialized type of process. However, it turns out that they are surprisingly general and all semimartingales which are absolutely continuous (in the appropriate sense) are Ito processes. This can be proven as a consequence of Lévy’s characterisation of Brownian motion.
Theorem 8 Let X be a continuous stochastic process, and suppose that there exists at least one Brownian motion defined on the underlying filtered probability space. Then the following are equivalent.
- X is a semimartingale such that for all bounded predictable processes satisfying .
- X satisfies decomposition (6) for some Brownian motion B and predictable processes satisfying
(almost surely) for each .
Proof: First, suppose that X satisfies decomposition (6) and that is a bounded predictable process with . Then, integrating with respect to shows that is zero. So, is a local martingale with quadratic variation , which is zero. Therefore, is constant and, hence, is zero.
Conversely, suppose that the first condition is satisfied. By Theorem 1, we can write for a continuous local martingale M and continuous FV process A. If is a bounded predictable process with then, as continuous FV processes have zero quadratic variation,
As previously shown in the post on Lévy’s characterization, this implies that there exists a Brownian motion B and a predictable process with (for all t) such that . It is here that we needed to assume the existence of at least one Brownian motion on the underlying filtered probability space.
Again, supposing that is a bounded predictable process such that then,
Integration with respect to Ito processes reduces to integration with respect to Brownian motion and Lebesgue-Stieltjes integration, by associativity of integration.
Theorem 9 Suppose that X satisfies decomposition (6). Then, a predictable process is X-integrable if and only if
(almost surely) for all times . In that case, the integral of with respect to X is given by
Proof: Writing and , Theorem 2 says that is X-integrable if and only if
as required. ⬜