In this post, I will give a statement and proof of the Bichteler-Dellacherie theorem describing the space of semimartingales. A semimartingale, as defined in these notes, is a cadlag adapted stochastic process X such that the stochastic integral is well-defined for all bounded predictable integrands . More precisely, an integral should exist which agrees with the explicit expression for elementary integrands, and satisfies bounded convergence in the following sense. If is a uniformly bounded sequence of predictable processes tending to a limit , then in probability as n goes to infinity. If such an integral exists, then it is uniquely defined up to zero probability sets.
An immediate consequence of bounded convergence is that the set of integrals for a fixed time t and bounded elementary integrands is bounded in probability. That is,
is bounded in probability, for each . For cadlag adapted processes, it was shown in a previous post that this is both a necessary and sufficient condition to be a semimartingale. Some authors use the property that (1) is bounded in probability as the definition of semimartingales (e.g., Protter, Stochastic Calculus and Differential Equations). The existence of the stochastic integral for arbitrary predictable integrands does not follow particularly easily from this definition, at least, not without using results on extensions of vector valued measures. On the other hand, if you are content to restrict to integrands which are left-continuous with right limits, the integral can be constructed very efficiently and, furthermore, such integrands are sufficient for many uses (integration by parts, Ito’s formula, a large class of stochastic differential equations, etc).
It was previously shown in these notes that, if X can be decomposed as for a local martingale M and FV process V then it is possible to construct the stochastic integral, so X is a semimartingale. The importance of the Bichteler-Dellacherie theorem is that it tells us that a process is a semimartingale if and only if it is the sum of a local martingale and an FV process. In fact this was the historical definition used of semimartingales, and is still probably the most common definition.
Throughout, we work with respect to a complete filtered probability space , and all processes are real-valued.
Theorem 1 (Bichteler-Dellacherie) For a cadlag adapted process X, the following are equivalent.
- X is a semimartingale.
- For each , the set given by (1) is bounded in probability.
- X is the sum of a local martingale and an FV process.
Furthermore, the local martingale term in 3 can be taken to be locally bounded.
So, the three alternative definitions of a semimartingale as a `good integrator’, a process for which (1) is bounded in probability, and as the sum of a local martingale and an FV process all agree with each other. The seemingly weakest of these three conditions is that (1) is bounded in probability, and that X is the sum of a local martingale and an FV process does seem like quite a strong condition. So, the main impact of the Bichteler-Dellacherie theorem is in the equivalence of these two conditions.
It is often useful to choose the martingale term in the semimartingale decomposition to be locally square integrable so that, for example, the Ito isometry can be used. This is always possible, and the proof given below naturally generates locally square integrable martingale terms. In fact, we have the stronger property that the martingale term is locally bounded. However, showing that is rather tricky and I will leave the proof until a later post.
As noted above, we have already shown that the first two conditions in the statement of the Bichteler-Dellacherie theorem are equivalent, and that the second and third both imply that X is a semimartingale. So, here, it is only necessary here to show that all semimartingales decompose into the sum of a local martingale and FV process. First, however, I will state and prove a useful consequence of the theorem. In many textbooks, stochastic integration with respect to a semimartingale X is constructed directly from the decomposition into local martingale and FV terms. This usually requires the integrand to be V-integrable in the Lebesgue-Stieltjes sense, so that is finite. The integral with respect to M is often constructed using the Ito isometry, and requires to be locally integrable. Alternatively, the Burkholder-Davis-Gundy inequality can be used, only requiring the weaker condition that is locally integrable. Then, a process is X-integrable if it is both M-integrable and V-integrable in the way just described. Note that there is a problem with this approach. The definition of X-integrability depends on the decomposition used, and this is not unique. Therefore, a process is said to be X-integrable if the exists any such decomposition with respect to which it is both M-integrable and V-integrable as just described. In these notes, on the other hand, the set of X-integrable processes was defined to be the largest possible class of predictable integrands with respect to which the dominated convergence theorem holds (i.e., they are good dominators). It is not easy to see that these two definitions lead to the same concept of X-integrability but, with the help of the Bichteler-Dellacherie theorem, it can be shown that this is indeed the case.
Theorem 2 If X is a semimartingale and is a predictable process, then the following are equivalent.
- is X-integrable.
- for some local martingale M and FV process V such that for each t (almost surely), and is locally integrable.
Then, is both M and V-integrable and
gives a decomposition of into the sum of a local martingale and an FV process.
Furthermore, it is possible to choose the decomposition in 2 so that M, and are locally bounded.
Proof: Suppose that condition 2 holds. Then, as previously shown, is V-integrable and the stochastic integral agrees with the Lebesgue-Stieltjes integral. In particular, is an FV process. Also, is M-integrable and is a local martingale. So, is X-integrable and (2) gives a decomposition of into the sum of a local martingale and an FV process.
Conversely, suppose that 1 holds and set , which is X-integrable. Then is a semimartingale and, by Theorem 1, decomposes as the sum of a local bounded martingale N and FV process W. As then is a local martingale with locally bounded jumps . So, M is locally bounded. Similarly, is an FV process. Associativity of the integral shows that and, using the fact that is bounded by 1,
As N was taken to be locally bounded, the jumps are locally bounded, so and are locally bounded. ⬜
Existence of the Decomposition
As mentioned above, the first two statements of 1 have already been shown to be equivalent earlier in these notes, and the third condition implies that X is a semimartingale. It remains to show that every semimartingale decomposes as the sum of a local martingale and an FV process, which I will do now. The idea is simple enough, and only really involves the existence of quadratic variations along with the integration by parts formula and some properties of martingales. To simplify things a bit, I will assume that the filtration is right-continuous in this section. However, all of the stated results do also hold in the non-right-continuous case, as I will show later.
The method of constructing the decomposition will be to orthogonally project the semimartingale into the space of cadlag martingales. We can start by defining a vector space as the set of semimartingales whose squared initial value and quadratic variation at infinity are integrable, . Then, the quadratic covariation enables us to define a symmetric bilinear form
As , the covariation is integrable for all , so this is well-defined. In fact, it defines a degenerate inner product. It only fails to be a true inner product because there exist non-constant semimartingales with zero quadratic variation (e.g., continuous FV processes). This does not matter here though.
Next, consider the space of cadlag and bounded martingales M, which I will denote by . By martingale convergence, we can define a map taking a martingale M to its limit at infinity . Furthermore, Ito’s isometry shows that is a subspace of and
So, Ito’s isometry gives us an isometry from to . Conversely, given any square-integrable random variable the martingale has a cadlag version (this requires right-continuity of the filtration). If U is -measurable then . So, is congruent to . In particular, this means that is a complete subspace of and, hence, there is an orthogonal projection from to . Writing for its orthogonal complement in , we have arrived at the following.
Lemma 3 Let X be a semimartingale such that is integrable. Then, there exists a unique decomposition for and .
We can describe the elements in a bit more detail. The constant process is a martingale, so and A almost surely starts from zero. Next, if is a bounded elementary predictable process and then is a martingale. Commuting the integral with the covariation,
So, is a martingale. Next, is stable under stopping at any stopping time ,
Localizing Lemma 3 gives the following. Recall that and denote the local martingales and local square integrable martingales respectively.
Lemma 4 Let X be a locally square integrable semimartingale. Then, it decomposes as where and A is such that for all .
Proof: Without loss of generality, we can suppose that . As X is locally square integrable, the same is true of its jumps . Then, is locally integrable, so is also locally integrable. Therefore, there exists a sequence stopping times increasing to infinity such that are integrable.
Apply Lemma 3 to obtain decompositions for and . For any we have . By uniqueness of the decomposition, this shows that , so for all . This means that we can define a process A by for all n and the local square integrable martingale M by . Then, .
Now, let . So, there is a sequence of stopping times increasing to infinity such that are bounded martingales. So, is a martingale and, therefore, is a local martingale. ⬜
We now move on to the main part of the proof of Theorem 1 and show that all semimartingales which are (locally) in the orthogonal complement are actually FV processes. The argument is rather tricky, but the basic idea is to compute the variation of A on an interval as the limit of the variation computed along a sequence of partitions. Then, on each partition, express the variation as a stochastic integral plus a martingale term. The stochastic integral term can be bounded using the fact that the set
is bounded in probability. This is a simple consequence of bounded convergence (see the proof of Lemma 3 of The Stochastic Integral). The martingale term can be bounded separately by taking expectations and, putting these together, gives a bound (in probability) on the variation of A.
Lemma 5 Let A be a locally integrable semimartingale such that for all cadlag bounded martingales M. Then, A is an FV process.
Proof: First, by localization, we can assume that is integrable. Then, we need to show that A almost surely has finite variation on each given time interval . The variation along a partition of the interval is defined as
If we were to take a sequence of partitions with mesh tending to zero and perform this calculation, then V would tend to the variation of A on . It just needs to be shown that, if we were to do this, then the sequence of random variables obtained for V is bounded in probability and, hence, can only converge to an almost surely finite limit. So, the problem reduces to bounding V in probability independently of the chosen partition. That is, fixing , we need to show that there is a positive real number K, not depending on the times , such that .
As the filtration is assumed to be right-continuous, it is possible to define cadlag martingales by
Note that this martingale is nonnegative, bounded by 1, and is constant over . Integration by parts gives the following expression for the absolute values of the increments of A across the intervals of the partition,
Summing this over i gives the variation along the partition as
where is a predictable process bounded by 1 and N is a local martingale given by,
The first term inside the parentheses defining N is a local martingale, as stochastic integration preserves the local martingale property, and the quadratic covariation terms are local martingales by the hypothesis of the lemma. For any time t in the size of the jump of N is . As M is bounded by 1 and A is bounded by , this gives .
Looking at (3), the first term on the right hand side is a stochastic integral with integrand bounded by 1, so is bounded in probability independently of (so, independently of the partition). To show that A has finite variation, it is enough to find an upper bound (in probability) for the martingale term . Choosing a positive real L, let be the first time t at which or . By the debut theorem, this is a stopping time. If is the smallest positive integer with then applying integration by parts and summing over i in the same way as above gives
Noting that the first term on the left hand side is nonnegative and the second is bounded by gives a lower bound for N,
Each of the terms on the right hand side are bounded in probability, independently of the choice of partition and the stopping time . So, choosing L large enough, we can ensure that . Again, L can be chosen independently of the partition. By construction, whenever , so
Next, as is bounded by , the stopped process is bounded below by the integrable random variable . Consequently, is a supermartingale so that and we can bound the expectation of its positive part,
Now, Markov’s inequality can be used to bound the probability that N is greater than any positive value K,
Taking gives as required, and this choice of K is independent of the partition. ⬜
Applying this result to Lemma 4, we see that every locally square integrable semimartingale decomposes as the sum of a local martingale and an FV process and, furthermore, that the terms in the decomposition can also be taken to be locally square integrable. All that remains is to extend this to all semimartingales, which is not difficult to do. Recalling that a cadlag adapted process X is locally bounded if and only if its jumps are locally bounded, simply subtract out all the jumps of X exceeding some fixed value. As stated in Lemma 6 below, this means that any semimartingale X decomposes as the sum of a locally bounded (and, in particular, locally square integrable) semimartingale Y and an FV process. So, applying the decomposition from Lemma 4 to Y gives a decomposition into a local martingale and an FV process.
Lemma 6 Let X be a cadlag adapted process. Then, it decomposes as where V is an FV process and Y satisfies . In particular, if X is a semimartingale then so is Y.
Proof: Take . As X is cadlag it can only have finitely many jumps of size greater than 1 in each bounded interval. So, V is an FV process and is a semimartingale. Therefore, has jumps , so is locally bounded, and is a semimartigale whenever X is. ⬜
This completes the proof of the Bichteler-Dellacherie theorem given here, at least for right-continuous filtrations. There is, still, one final point to add. Theorem 1 contains the additional claim that the martingale term in the decomposition can be taken to be locally bounded. The construction above already gives us a locally square integrable martingale term, which is strong enough for most applications. However, as mentioned above, the proof of the stronger locally bounded property will be left until a later post. In fact, it will follow from properties of predictable processes that the martingale term constructed above is already locally bounded. Actually, as well as being an FV process, any A satisfying the conditions of Lemma 5 is also predictable which, as will be shown, implies that it is locally bounded. So, Lemma 4 decomposes a locally bounded semimartingale into the sum of a locally bounded FV process and a local martingale which, being the difference of two locally bounded processes, must itself be locally bounded. Applying this to the semimartingale Y from Lemma 6 automatically results in a locally bounded martingale term.
The Bichteler-Dellacherie theorem has been proven above in the case where the filtration is right-continuous, so that is equal to for all times t. Being one of the usual conditions, right-continuity is often assumed to hold in stochastic process theory. However, the conclusion of the theorem does still hold without imposing this condition, and I will show why this is the case here. While it is possible to drop this condition from the start, assuming right-continuity does simplify the argument, which is why it was assumed it above.
Passing between the original filtration and the right-continuous one can be done without too much trouble. For example, consider the space of -predictable processes. By definition, this is generated by the left-continuous and -adapted processes but, using left-continuity, any such process is automatically -adapted at all positive times t. Therefore, is -predictable. This means that if X is a semimartingale with respect to the original filtration, then the stochastic integral is well-defined for all -predictable processes . By bounded convergence, it is easily checked that this also agrees with the explicit expression for all -elementary integrands. So, by the definition used in these notes, X is also an -semimartingale. Conversely, if X is an -semimartingale and is adapted under the original filtration then, as the stochastic integral agrees by definition with the expression for elementary integrands, it is clearly also a semimartingale under the original filtration.
Lemma 7 An adapted process X is an -semimartingale if and only if it is an -semimartingale.
Now, suppose that X is a semimartingale under the original filtration such that is integrable. I’ll write for the space of -bounded and cadlag -martingales. Then, applying Lemma 3 under gives a decomposition where and A is an -semimartingale satisfying and for all . In fact, it can be shown that A is -adapted by applying Lemma 8 below under the filtration . So, , and Lemma 3 generalises to the non-right-continuous case. Alternatively, Doob’s inequality can be used to show that any limit of cadlag martingales in the sense also has a cadlag modification. Then, is seen to be complete, and the proof of Lemma 3 follows in the same way as above, without making any reference to right-continuity of the filtration.
Lemma 8 Suppose that A is a semimartingale such that is integrable and for all uniformly bounded cadlag martingales M. Then, is -measurable for all positive times t.
Proof: Suppose that U is a bounded -measurable random variable. Then, is a martingale. So, is zero and, rearranging this gives . So, and, hence, are -measurable. ⬜
The only remaining place where right-continuity was required above was in the proof of Lemma 5. In fact, if A satisfies the conditions of that Lemma, then it also satisfies the condition under the right-continuous filtration . Then, the conclusion that A has finite variation over all bounded time intervals will hold (and this does not refer to any filtration). To pass to the right-continuous filtration, the following simple Lemma allows us to convert -martingales to -martingales.
Lemma 9 Suppose that M is a cadlag and square-integrable -martingale. Then, there exists a countable subset such that for all . Furthermore, is an -martingale.
Proof: As it is cadlag, there can only be finitely many times at which on a bounded time interval [0,T] for each . So, there are only finitely many such times at which . Then, letting decrease to zero and T increase to infinity, there is only a countable set of times S at which .
Now, suppose that S is as above, and set . This is a square-integrable martingale with respect to . To show that it is also an -martingale, it is enough to show that (almost surely) for each fixed time t, so that it is adapted. However, . If then this is zero and, if then so, again, . ⬜
Now, suppose that A satisfies the conditions of Lemma 5 and M is a cadlag bounded -martingale. By localization, it can be assumed that is integrable. Then, let be a sequence of distinct times including the set S of all t for which . Lemma 9 above shows that is a martingale under the original filtration. The quadratic covariation can be decomposed as
The term is a local martingale under the original filtration, by assumption on the properties of A, so it is also an -local martingale. As is -measurable (Lemma 8), the terms inside the summation are also -martingales. So, by dominated convergence of the sum, we see that is an -local martingale, and the conditions of Lemma 8 also hold with respect to this right-continuous filtration.
I will give a few further comments regarding the proof the Bichteler-Dellacherie theorem given here. It is instructive, I think, to compare this method with the more common approach found in many textbooks. There, the approach is to reduce the problem to the case of quasimartingales. These are cadlag adapted and integrable processes such that the set
is bounded. In that case, a generalization of the Doob-Meyer decomposition to quasimartingales says that X decomposes as for a martingale M and integrable variation process A. The method of reducing to quasimartingales is to consider the set defined in (1) above, for any fixed time t. This is bounded in probability, and, by the Hahn-Banach theorem, it is possible to show that for any convex which is bounded in probability, there exists a uniformly bounded and strictly positive random variable Z such that is bounded. Normalising Z so that , this can be used to show that X is a quasimartingale under the equivalent measure . Equivalently, defining the martingale , the process is a quasimartingale, so decomposes as the sum of a martingale M and an FV process by Rao’s decomposition. Then, Ito’s formula can be used to express X as the local martingale plus an FV process.
On the other hand, the approach used here is via Lemma 3 above, which decomposes X uniquely into a martingale term plus an orthogonal component. This is quite direct, and the main work involved is in showing that the orthogonal component does have finite variation (Lemma 5). One advantage of this approach is that it avoids the use of the Hahn-Banach theorem and the associated arbitrary change of measure (or auxiliary martingale Z). It also results in a decomposition which is essentially unique. In fact, the decomposition in Lemma 4 for locally square integrable semimartingales is unique, and the general case gives a unique decomposition after all large jumps have been subtracted out. This uniqueness is expressed in the canonical decomposition of special semimartingales, which I will cover in a later post.
It is also interesting to ask how much stochastic calculus is really required in the proof above. The existence of cadlag versions of martingales was required, as was the existence of quadratic variations, the integration by parts formula, and the Ito isometry. No other big results were needed. I did also make use of the equivalence between the first two statements of Theorem 1 right from the start of the post though. That is, if the set defined by (1) is bounded in probability, then it is possible to define the stochastic integral for bounded predictable . This is rather tricky to prove, and was assumed for convenience, since it has already been proved in an earlier post. However, this is not strictly necessary. Just assuming that the second statement of the theorem holds then it is easy to define the integral for integrands which are adapted, left-continuous, and right-continuous in probability and satisfying a rather weak uniform convergence property. That is, if uniformly then in probability. This is enough to prove results such as integration by parts and the Ito isometry for such integrands, and is enough to be able to apply the proof given here to decompose X as the sum of a local martingale and an FV process. See Protter, Stochastic Calculus and Differential Equations, for a development of stochastic integration in this way. However, even in Protter, the construction of the stochastic integral for arbitrary predictable integrands does depend on the standard proof of the Bichteler-Dellacherie theorem as just outlined.
Finally, consider the proof that condition 3 of the Bichteler-Dellacherie theorem as stated above implies that X is a semimartingale, so that the stochastic integral exists. Here, I referred to the proof given earlier in these notes which applies to all local martingales. However, the proof of condition 3 above gives a locally square integrable martingale term in the decomposition of X. In this case, the standard construction via the Ito isometry is quite efficient. By localising, and ignoring the finite variation term (which is clearly a semimartingale), it can be assumed that X is a square-integrable martingale. Applying the Ito isometry for elementary integrands gives
for elementary . However, the right hand side is well-defined for all bounded (by Lebesgue-Stieltjes integration). For bounded predictable integrands, the stochastic integral can be defined as the unique linear extension such that this identity still holds.