A major foundational result in stochastic calculus is that integration can be performed with respect to any local martingale. In these notes, a semimartingale was defined to be a cadlag adapted process with respect to which a stochastic integral exists satisfying some simple desired properties. Namely, the integral must agree with the explicit formula for elementary integrands and satisfy bounded convergence in probability. Then, the existence of integrals with respect to local martingales can be stated as follows.
Theorem 1 Every local martingale is a semimartingale.
This result can be combined directly with the fact that FV processes are semimartingales.
Corollary 2 Every process of the form X=M+V for a local martingale M and FV process V is a semimartingale.
Working from the classical definition of semimartingales as sums of local martingales and FV processes, the statements of Theorem 1 and Corollary 2 would be tautologies. Then, the aim of this post is to show that stochastic integration is well defined for all classical semimartingales. Put in another way, Corollary 2 is equivalent to the statement that classical semimartingales satisfy the semimartingale definition used in these notes. The converse statement will be proven in a later post on the Bichteler-Dellacherie theorem, so the two semimartingale definitions do indeed agree.
Let us now move on to the proof of Theorem 1 noting that, by localization, it is sufficient to prove the result for proper martingales. The previous post on existence of stochastic integrals will be used. To show that a cadlag martingale X is a semimartingale, we need to prove that the following set of elementary integrals is bounded in probability for each positive time t.
This is equivalent to stating that there is a function such that f(K) tends to zero as K goes to infinity and,
for all elementary predictable . As always in these notes, we work with respect to a complete filtered probability space . Then, on any interval (0,t], an elementary process can be written in the form
for times and -measurable random variables . For such integrands, the stochastic integral is
The proof that X is is semimartingale is easiest for square integrable martingales, so we handle that case first before generalizing to arbitrary martingales.
Square Integrable Martingales
The aim for now is to prove the following result for square integrable martingales.
Lemma 3 Let X be a square integrable martingale and be elementary predictable. Then,
For the remainder of this section, assume that is as in (2). Restricting to the finite set of times gives the discrete filtration , reducing the problem to one involving only discrete-time stochastic processes. Given a discrete-time process denote its increments , quadratic variation [Y] and `integral’ (for a discrete-time integrand Z) as follows,
The identity follows immediately from this. Summing over k gives the following discrete integration by parts formula
where denotes the process . If, furthermore, Y is a discrete square integrable martingale, so that , then taking expectations gives
Note also that if Y is a martingale and Z is a bounded and `discrete predictable’ process (i.e, is -measurable), then . So, will also be a martingale, and can be used in the place of Y in (6) to obtain the discrete version of Ito’s isometry,
which holds for square integrable martingales Y and bounded predictable Z. Finally, if then Ito’s isometry gives the inequality
Lemma 4 There exists a constant c such that the following inequality holds for any martingale X, elementary and positive real number K.
Taking in (1), this result shows that cadlag martingales are indeed semimartingales. It was shown by Burkholder that the optimal value for c is 2 (Sharp norm comparison of martingale maximal functions and stochastic integrals). However, for the purposes of proving Theorem 1 it is only necessary to prove the result for some c, and we do not attempt to find the best value.
The idea behind the proof of the lemma is to stop the martingale X before it gets too large, and then the previous method for square integrable martingales can be applied. It helps to first choose any time . Then, define the stopping time
Doob’s inequality bounds the probability that the maximum of X exceeds K,
Next, define the discrete process Y to be equal to X prior to and 0 afterwards
By definition, this is uniformly bounded by K and . Also, letting Z be as in (2) and taking gives
whenever . Unfortunately, the process Y is not a martingale. However, it can be approximated by the stopped process , which satisfies . Also, by the martingale property, for any discrete, predictable and bounded process H. If, furthermore, then, letting m be such that ,
Now, a Doob decomposition is used, where M is a discrete martingale and A is a discrete predictable process, both starting at 0. The processes M and A are defined by
The following identity follows from this definition,
Taking expectations and summing over k,
Then applying (7) gives the following bound for the integral with respect to M,
This makes use of the integration by parts formula (5) and, as , the inequality is used. The final inequality comes from (10) with in place of H. So, Chebyshev’s inequality can be used to bound the probability
Let us now look at the integral with respect to A. Defining gives and the following is obtained,
using (10) for the final inequality. So, applying Chebyshev’s inequality one more time,
Combining this with (11) ,
It is also possible to construct the integral with respect to any cadlag martingale without relying on the previous post on existence of stochastic integrals. This isn’t any less work than the proof used above though.
First, for a square integrable and cadlag martingale X, define the linear function on the space of bounded elementary processes,
As is a submartingale, this is a positive map. The following identity is straightforward,
Furthermore, using the fact that positive submartingales are of class (DL), it can be shown that is countably additive on the nonnegative elementary processes. Then, Carathéodory’s extension theorem allows to be extended to a measure on , which is known as the Doléans measure of . So, for each positive time t, defines an inner product on the space of bounded predictable processes and equation (12) says that the stochastic integral is an isometry from the elementary processes to . It can then be uniquely extended as an isometry from the space of bounded predictable processes.
The extension to an arbitrary cadlag martingale X can be achieved by approximating by a sequence of square integrable martingales, so that . Lemma 4 implies that convergence also holds under the semimartingale topology so that, by completeness, X will also be a semimartingale.
Compare this approach to the classical proof, which uses the Ito isometry for square integrable martingales
where is an increasing process such that is a martingale. The extension to arbitrary local martingales involves a decomposition X=M+V, for a locally square integrable martingale M and FV process V. The method outlined here with the Doléans measure avoids the rather involved construction of and decompositions of martingales. The proof given in the main body of this post also avoids having to construct the Doléans measure, instead relying on a characterization of semimartingales in terms of boundedness in probability.